8.3. Integration by Parts and Substitution Formula


In this part we will restate two essential derivation rules, the product and the chain rule, in their integral shape. The resulting rules, called integration by parts and substitution formula, are valuable tools in integral calculus.

As before I still denotes an arbitrary interval.

Theorem (integration by parts):  Let f and g be two differentiable functions on I, i.e f,g D 1 (I) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacYcacaWGNbGaeyicI4SaamiramaaCaaaleqabaGaaGymaaaakiaacIcacaWGjbGaaiykaaaa@3DD9@ . Then the following holds:

If f g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaeyyXICTaam4zaaaa@3A19@ is integrable on I then f g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabgwSixlqadEgagaqbaaaa@3A19@ is integrable as well and for all a,bI MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacYcacaWGIbGaeyicI4Saamysaaaa@3ABB@ we have:

a b f g =fg | a b a b f g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacaWGMbGaeyyXICTabm4zayaafaaaleaacaWGHbaabaGaamOyaaqdcqGHRiI8aOGaeyypa0JaamOzaiabgwSixlaadEgacaGG8bWaa0baaSqaaiaadggaaeaacaWGIbaaaOGaeyOeI0Yaa8qCaeaaceWGMbGbauaacqGHflY1caWGNbaaleaacaWGHbaabaGaamOyaaqdcqGHRiI8aaaa@4FD4@
[8.3.1]

Proof:  With the product rule [7.7.6] we find that fg MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabgwSixlaadEgaaaa@3A0D@ is differentiable on I and that (fg ) = f g+f g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAgacqGHflY1caWGNbGabiykayaafaGaeyypa0JabmOzayaafaGaeyyXICTaam4zaiabgUcaRiaadAgacqGHflY1ceWGNbGbauaaaaa@45B4@ . In other words: f g+f g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaeyyXICTaam4zaiabgUcaRiaadAgacqGHflY1ceWGNbGbauaaaaa@3F28@ has a primitive, namely fg MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabgwSixlaadEgaaaa@3A0D@ . Now, if f g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaeyyXICTaam4zaaaa@3A19@ is integrable the same is true for f g = f g+f g f g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabgwSixlqadEgagaqbaiabg2da9iqadAgagaqbaiabgwSixlaadEgacqGHRaWkcaWGMbGaeyyXICTabm4zayaafaGaeyOeI0IabmOzayaafaGaeyyXICTaam4zaaaa@4975@ due to [8.1.7] and eventually the identity

a b f g + a b f g = a b f g+f g =fg | a b MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaaceWGMbGbauaacqGHflY1caWGNbaaleaacaWGHbaabaGaamOyaaqdcqGHRiI8aOGaey4kaSYaa8qCaeaacaWGMbGaeyyXICTabm4zayaafaaaleaacaWGHbaabaGaamOyaaqdcqGHRiI8aOGaeyypa0Zaa8qCaeaaceWGMbGbauaacqGHflY1caWGNbGaey4kaSIaamOzaiabgwSixlqadEgagaqbaaWcbaGaamyyaaqaaiaadkgaa0Gaey4kIipakiabg2da9iaadAgacqGHflY1caWGNbGaaiiFamaaDaaaleaacaWGHbaabaGaamOyaaaaaaa@5E4B@

yields the assertion [8.3.1].

Consider:

  • If, in addition, f is even continuously differentiable, i.e. f C 1 (I) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabgIGiolaadoeadaahaaWcbeqaaiaaigdaaaGccaGGOaGaamysaiaacMcaaaa@3C3C@ , then f g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaeyyXICTaam4zaaaa@3A19@ is continuous on I and thus integrable as well.

  • f and g play symmetric roles, so that integration by parts might well be quoted as

    a b f g =fg | a b a b f g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaaceWGMbGbauaacqGHflY1caWGNbaaleaacaWGHbaabaGaamOyaaqdcqGHRiI8aOGaeyypa0JaamOzaiabgwSixlaadEgacaGG8bWaa0baaSqaaiaadggaaeaacaWGIbaaaOGaeyOeI0Yaa8qCaeaacaWGMbGaeyyXICTabm4zayaafaaaleaacaWGHbaabaGaamOyaaqdcqGHRiI8aaaa@4FD4@
     
  • Using this rule however requires to go for only one variant. Though both alternatives are correct, in most cases there is only one wise option to choose. For a "safe" handling some experience is essential.

  • Integration by parts is only applicable if the integrand is a product with at least one factor having a known primitive.
    This explains the rule's name: We don't need to integrate (i.e. to find a primitive for) the whole of the integrand, but it is sufficient to integrate only a part of it.

     

Integration by parts is mainly used for calculating primitives according to the fundamental theorem [8.2.13]. Some examples will explain this technique.

Example:  

  • Xcos+sin MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaamiwaiabgwSixlGacogacaGGVbGaai4CaiabgUcaRiGacohacaGGPbGaaiOBaaaa@408D@ is a primitive function of Xsin MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwaiabgwSixlGacohacaGGPbGaaiOBaaaa@3BEB@ as all x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolabl2riHcaa@39DD@ satisfy

    0 x Xsin = 0 x X(cos ) =X(cos) | 0 x 0 x X (cos) =Xcos | 0 x + 0 x cos =Xcos | 0 x +sin | 0 x =xcosx+sinx MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@993E@

     
  • X 2 cos+2Xsin+2cos MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaamiwamaaCaaaleqabaGaaGOmaaaakiabgwSixlGacogacaGGVbGaai4CaiabgUcaRiaaikdacaWGybGaeyyXICTaci4CaiaacMgacaGGUbGaey4kaSIaaGOmaiGacogacaGGVbGaai4Caaaa@49D4@ is a primitive function of X 2 sin MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaCaaaleqabaGaaGOmaaaakiabgwSixlGacohacaGGPbGaaiOBaaaa@3CDE@ : We integrate by parts twice and get for x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolabl2riHcaa@39DD@

    0 x X 2 sin = 0 x X 2 (cos ) = X 2 (cos) | 0 x + 0 x 2Xcos = X 2 cos | 0 x + 0 x 2Xsin = X 2 cos | 0 x +2Xsin | 0 x 0 x 2sin = X 2 cos | 0 x +2Xsin | 0 x +2cos | 0 x = x 2 cosx+2xsinx+2cosx2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@DA49@

    and thus know that X 2 cos+2Xsin+2cos2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaamiwamaaCaaaleqabaGaaGOmaaaakiabgwSixlGacogacaGGVbGaai4CaiabgUcaRiaaikdacaWGybGaeyyXICTaci4CaiaacMgacaGGUbGaey4kaSIaaGOmaiGacogacaGGVbGaai4CaiabgkHiTiaaikdaaaa@4B7D@ is a primitive of X 2 sin MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaCaaaleqabaGaaGOmaaaakiabgwSixlGacohacaGGPbGaaiOBaaaa@3CDE@ . Finally we omit the constant addend −2.

  • The last example reveals a primitive of cos 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbWaaWbaaSqabeaacaaIYaaaaaaa@39A8@ . The calculation uses Pythagoras' theorem (i.e. the identity sin 2 =1 cos 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacMgacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaeyypa0JaaGymaiabgkHiTiGacogacaGGVbGaai4CamaaCaaaleqabaGaaGOmaaaaaaa@4021@ ), a standard trick!

    At first we have for x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolabl2riHcaa@39DD@ :

    0 x cos 2 = 0 x coscos = 0 x cossin =cossin | 0 x 0 x cos sin =cossin | 0 x + 0 x sin 2 =cossin | 0 x + 0 x 1 cos 2 =cossin | 0 x + 0 x 1 0 x cos 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@B94A@

    Indeed this is not the expected solution for our integral! Instead however it is an equation satisfied by the unknown integral that could be solved for it:

    0 x cos 2 = 1 2 (sincos | 0 x + 0 x 1 )= 1 2 (sinxcosx+x) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaaciGGJbGaai4BaiaacohadaahaaWcbeqaaiaaikdaaaaabaGaaGimaaqaaiaadIhaa0Gaey4kIipakiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaGaaiikaiGacohacaGGPbGaaiOBaiabgwSixlGacogacaGGVbGaai4CaiaacYhadaqhaaWcbaGaaGimaaqaaiaadIhaaaGccqGHRaWkdaWdXbqaaiaaigdaaSqaaiaaicdaaeaacaWG4baaniabgUIiYdGccaGGPaGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaacaGGOaGaci4CaiaacMgacaGGUbGaamiEaiabgwSixlGacogacaGGVbGaai4CaiaadIhacqGHRaWkcaWG4bGaaiykaaaa@620B@

    Finally this proves 1 2 (sincos+X) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaaGOmaaaacaGGOaGaci4CaiaacMgacaGGUbGaeyyXICTaci4yaiaac+gacaGGZbGaey4kaSIaamiwaiaacMcaaaa@4280@ as a primitive of cos 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbWaaWbaaSqabeaacaaIYaaaaaaa@39A8@ .

The Pythagorean theorem is often involved when integrating by parts. As another example we prove the recursion formulas for the integrals of sin n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacMgacaGGUbWaaWbaaSqabeaacaWGUbaaaaaa@39E4@ and cos n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbWaaWbaaSqabeaacaWGUbaaaaaa@39DF@ .

Proposition:  For all a,b MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacYcacaWGIbGaeyicI4SaeSyhHekaaa@3B5D@ and all n2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgwMiZkaaikdaaaa@3961@ the following recursions are valid:

  1. a b sin n = cos sin n1 n | a b + n1 n a b sin n2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaaciGGZbGaaiyAaiaac6gadaahaaWcbeqaaiaad6gaaaaabaGaamyyaaqaaiaadkgaa0Gaey4kIipakiabg2da9iabgkHiTmaalaaabaGaci4yaiaac+gacaGGZbGaeyyXICTaci4CaiaacMgacaGGUbWaaWbaaSqabeaacaWGUbGaeyOeI0IaaGymaaaaaOqaaiaad6gaaaGaaiiFamaaDaaaleaacaWGHbaabaGaamOyaaaakiabgUcaRmaalaaabaGaamOBaiabgkHiTiaaigdaaeaacaWGUbaaamaapehabaGaci4CaiaacMgacaGGUbWaaWbaaSqabeaacaWGUbGaeyOeI0IaaGOmaaaaaeaacaWGHbaabaGaamOyaaqdcqGHRiI8aaaa@5D26@

[8.3.2]
  1. a b cos n = sin cos n1 n | a b + n1 n a b cos n2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaaciGGJbGaai4BaiaacohadaahaaWcbeqaaiaad6gaaaaabaGaamyyaaqaaiaadkgaa0Gaey4kIipakiabg2da9maalaaabaGaci4CaiaacMgacaGGUbGaeyyXICTaci4yaiaac+gacaGGZbWaaWbaaSqabeaacaWGUbGaeyOeI0IaaGymaaaaaOqaaiaad6gaaaGaaiiFamaaDaaaleaacaWGHbaabaGaamOyaaaakiabgUcaRmaalaaabaGaamOBaiabgkHiTiaaigdaaeaacaWGUbaaamaapehabaGaci4yaiaac+gacaGGZbWaaWbaaSqabeaacaWGUbGaeyOeI0IaaGOmaaaaaeaacaWGHbaabaGaamOyaaqdcqGHRiI8aaaa@5C2F@

[8.3.3]

Proof:  Verification is quite similar in both cases, so it is sufficient to prove only one of them, e.g. 2. From sin 2 =1 cos 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacMgacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaeyypa0JaaGymaiabgkHiTiGacogacaGGVbGaai4CamaaCaaaleqabaGaaGOmaaaaaaa@4021@ we get the following equation for a b cos n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaaciGGJbGaai4BaiaacohadaahaaWcbeqaaiaad6gaaaaabaGaamyyaaqaaiaadkgaa0Gaey4kIipaaaa@3E0A@ :

a b cos n = a b sin cos n1 =sin cos n1 | a b a b sin(n1) cos n2 (sin) =sin cos n1 | a b +(n1) a b sin 2 cos n2 =sin cos n1 | a b +(n1) a b (1 cos 2 ) cos n2 =sin cos n1 | a b +(n1) a b cos n2 (n1) a b cos n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@EE35@

Thus we have:  n a b cos n =sin cos n1 | a b +(n1) a b cos n2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaapehabaGaci4yaiaac+gacaGGZbWaaWbaaSqabeaacaWGUbaaaaqaaiaadggaaeaacaWGIbaaniabgUIiYdGccqGH9aqpciGGZbGaaiyAaiaac6gacqGHflY1ciGGJbGaai4BaiaacohadaahaaWcbeqaaiaad6gacqGHsislcaaIXaaaaOGaaiiFamaaDaaaleaacaWGHbaabaGaamOyaaaakiabgUcaRiaacIcacaWGUbGaeyOeI0IaaGymaiaacMcadaWdXbqaaiGacogacaGGVbGaai4CamaaCaaaleqabaGaamOBaiabgkHiTiaaikdaaaaabaGaamyyaaqaaiaadkgaa0Gaey4kIipaaaa@5C75@ which in fact is the assertion.

If a and b are zeros for the sine or for the cosine the recursion formulas could be simplified to

a b sin n = n1 n a b sin n2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaaciGGZbGaaiyAaiaac6gadaahaaWcbeqaaiaad6gaaaaabaGaamyyaaqaaiaadkgaa0Gaey4kIipakiabg2da9maalaaabaGaamOBaiabgkHiTiaaigdaaeaacaWGUbaaamaapehabaGaci4CaiaacMgacaGGUbWaaWbaaSqabeaacaWGUbGaeyOeI0IaaGOmaaaaaeaacaWGHbaabaGaamOyaaqdcqGHRiI8aaaa@4C89@    and    a b cos n = n1 n a b cos n2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaaciGGJbGaai4BaiaacohadaahaaWcbeqaaiaad6gaaaaabaGaamyyaaqaaiaadkgaa0Gaey4kIipakiabg2da9maalaaabaGaamOBaiabgkHiTiaaigdaaeaacaWGUbaaamaapehabaGaci4yaiaac+gacaGGZbWaaWbaaSqabeaacaWGUbGaeyOeI0IaaGOmaaaaaeaacaWGHbaabaGaamOyaaqdcqGHRiI8aaaa@4C7F@ .

In this case we also succeed in finding a non recursive representation. The following integral is needed in [8.5.7] where we will calculate the volume of a sphere.

Proposition:  For all n0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgwMiZkaaicdaaaa@395F@ we have

π 2 π 2 cos n = { n! ( 2 k k!) 2 π   if  n=2k ( 2 k k!) 2 n! 2   if  n=2k+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6C80@
[8.3.4]

Proof:  The identity [8.3.4] is immediate if k=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiabg2da9iaaicdaaaa@389C@ . Without restriction we thus assume that k>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiabg6da+iaaicdaaaa@389E@ . If n=2k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabg2da9iaaikdacaWGRbaaaa@3991@ the simplified recursion formula could be applied exactly k times:

π 2 π 2 cos n = n1 n n3 n2 1 2 π 2 π 2 cos 0 = n(n1) n 2 (n2)(n3) (n2) 2 21 2 2 π = n! ((2k)(2k2)(2k2(k1))) 2 π = n! ( 2 k (k(k1)(k(k1))) 2 π (2 factored out k times) = n! ( 2 k k!) 2 π MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@E96B@

We proceed similar in the case n=2k+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabg2da9iaaikdacaWGRbGaey4kaSIaaGymaaaa@3B2E@ and again apply the simplified recursion formula k times:

π 2 π 2 cos n = n1 n n3 n2 2 3 π 2 π 2 cos 1 = (n1) 2 n(n1) (n3) 2 (n2)(n3) 2 2 32 2 = ((2k)(2k2)(2k2(k1))) 2 n! 2 = ( 2 k (k(k1)(k(k1))) 2 n! 2 = ( 2 k k!) 2 n! 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@D941@

We now turn to the chain rule and its integral version. Different from integration by parts the substitution formula also controls the bounderies of integration.

Theorem (substitution formula):  Let I and J be any two intervals and g:JI MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiaacQdacaWGkbGaeyOKH4Qaamysaaaa@3B20@ a differentiable function. If f is integrable on I, i.e. fI(I) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabgIGiolaadMeacaGGOaGaamysaiaacMcaaaa@3B50@ then (fg) g I(J) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAgacqWIyiYBcaWGNbGaaiykaiabgwSixlqadEgagaqbaiabgIGiolaadMeacaGGOaGaamOsaiaacMcaaaa@4212@ and for all a,bJ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacYcacaWGIbGaeyicI4SaamOsaaaa@3ABC@ the following identity holds:

a b (fg) g = g(a) g(b) f MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacaGGOaGaamOzaiablIHiVjaadEgacaGGPaGaeyyXICTabm4zayaafaaaleaacaWGHbaabaGaamOyaaqdcqGHRiI8aOGaeyypa0Zaa8qCaeaacaWGMbaaleaacaWGNbGaaiikaiaadggacaGGPaaabaGaam4zaiaacIcacaWGIbGaaiykaaqdcqGHRiI8aaaa@4C89@
[8.3.5]

Proof:  Let h be a primitive function of f. According to the chain rule ([7.7.8]) the composit hg MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAaiablIHiVjaadEgaaaa@38FF@ is differentiable on I and

(hg ) =( h g) g =(fg) g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadIgacqWIyiYBcaWGNbGabiykayaafaGaeyypa0JaaiikaiqadIgagaqbaiablIHiVjaadEgacaGGPaGaeyyXICTabm4zayaafaGaeyypa0JaaiikaiaadAgacqWIyiYBcaWGNbGaaiykaiabgwSixlqadEgagaqbaaaa@4BD6@ .

Thus (fg) g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAgacqWIyiYBcaWGNbGaaiykaiabgwSixlqadEgagaqbaaaa@3D98@ has hg MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAaiablIHiVjaadEgaaaa@38FF@ as a primitive and therefore

a b (fg) g =(hg) | a b =h | g(a) g(b) = g(a) g(b) f MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacaGGOaGaamOzaiablIHiVjaadEgacaGGPaGaeyyXICTabm4zayaafaaaleaacaWGHbaabaGaamOyaaqdcqGHRiI8aOGaeyypa0JaaiikaiaadIgacqWIyiYBcaWGNbGaaiykaiaacYhadaqhaaWcbaGaamyyaaqaaiaadkgaaaGccqGH9aqpcaWGObGaaiiFamaaDaaaleaacaWGNbGaaiikaiaadggacaGGPaaabaGaam4zaiaacIcacaWGIbGaaiykaaaakiabg2da9maapehabaGaamOzaaWcbaGaam4zaiaacIcacaWGHbGaaiykaaqaaiaadEgacaGGOaGaamOyaiaacMcaa0Gaey4kIipaaaa@5E80@

Consider:

  • Especially with the substitution formula the dx-notation is very common. In this world however it is not only the identity's differential dx that has to be considered but also the differential

    dg(x)= g (x)dx MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaadEgacaGGOaGaamiEaiaacMcacqGH9aqpceWGNbGbauaacaGGOaGaamiEaiaacMcacqGHflY1caWGKbGaamiEaaaa@429B@  

     i

    We supplement the annotations on differential forms of degree 1 in 8.2 and take up the identity d x X=X MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBaaaleaacaWG4baabeaakiaadIfacqGH9aqpcaWGybaaaa@3AC8@ . For r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabgIGiolabl2riHcaa@39D7@ we then have

    d x g(r)= g (x)r= g (x) d x X(r) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBaaaleaacaWG4baabeaakiaadEgacaGGOaGaamOCaiaacMcacqGH9aqpceWGNbGbauaacaGGOaGaamiEaiaacMcacqGHflY1caWGYbGaeyypa0Jabm4zayaafaGaaiikaiaadIhacaGGPaGaeyyXICTaamizamaaBaaaleaacaWG4baabeaakiaadIfacaGGOaGaamOCaiaacMcaaaa@4EC0@ ,

    i.e. (x, d x g)=(x, g (x) d x X) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadIhacaGGSaGaamizamaaBaaaleaacaWG4baabeaakiaadEgacaGGPaGaeyypa0JaaiikaiaadIhacaGGSaGabm4zayaafaGaaiikaiaadIhacaGGPaGaeyyXICTaamizamaaBaaaleaacaWG4baabeaakiaadIfacaGGPaaaaa@4897@ , which proves dg= g dX MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaadEgacqGH9aqpceWGNbGbauaacqGHflY1caWGKbGaamiwaaaa@3DCF@ often denoted as

    dg(x)= g (x)dx MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaadEgacaGGOaGaamiEaiaacMcacqGH9aqpceWGNbGbauaacaGGOaGaamiEaiaacMcacqGHflY1caWGKbGaamiEaaaa@429B@

    in the substitution formula's context.

    of an arbitrary differentiable function g.

    Now, if we substitute t=g(x) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabg2da9iaadEgacaGGOaGaamiEaiaacMcaaaa@3B2D@ , and consequently dt= g (x)dx MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaadshacqGH9aqpceWGNbGbauaacaGGOaGaamiEaiaacMcacaWGKbGaamiEaaaa@3E08@ we simply get the identity

    a b f(g(x)) g (x)dx = g(a) g(b) f(t)dt MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacaWGMbGaaiikaiaadEgacaGGOaGaamiEaiaacMcacaGGPaGaeyyXICTabm4zayaafaGaaiikaiaadIhacaGGPaGaamizaiaadIhaaSqaaiaadggaaeaacaWGIbaaniabgUIiYdGccqGH9aqpdaWdXbqaaiaadAgacaGGOaGaamiDaiaacMcacaWGKbGaamiDaaWcbaGaam4zaiaacIcacaWGHbGaaiykaaqaaiaadEgacaGGOaGaamOyaiaacMcaa0Gaey4kIipaaaa@5615@

    and it is is a valid one, guaranteed by the substitution formula! For the readers convenience the examples to follow are displayable either way. Just click the buttons and to choose your option.

  • The substitution formula is bidirectional. We use the left to right direction if the integrand is clearly shaped like (fg) g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAgacqWIyiYBcaWGNbGaaiykaiabgwSixlqadEgagaqbaaaa@3D98@ . In this case the substituition g is simply read off.

    With the second direction (right to left) we have to introduce independently a substitution g such that the integral of (fg) g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAgacqWIyiYBcaWGNbGaaiykaiabgwSixlqadEgagaqbaaaa@3D98@ is easier to calculate than the one of f. Further we now need to find inverse images of the boundaries with respect to g. If g is bijective we can employ the inverse function of g and restate [8.3.5] for a,bI MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacYcacaWGIbGaeyicI4Saamysaaaa@3ABB@ as follows:

    a b f = g 1 (a) g 1 (b) (fg) g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacaWGMbaaleaacaWGHbaabaGaamOyaaqdcqGHRiI8aOGaeyypa0Zaa8qCaeaacaGGOaGaamOzaiablIHiVjaadEgacaGGPaGaeyyXICTabm4zayaafaaaleaacaWGNbWaaWbaaWqabeaacqGHsislcaaIXaaaaSGaaiikaiaadggacaGGPaaabaGaam4zamaaCaaameqabaGaeyOeI0IaaGymaaaaliaacIcacaWGIbGaaiykaaqdcqGHRiI8aaaa@504B@
     

We start practising with two examples for the left to right direction.

Example:  We calculate the integral   0 1 ( X 2 +1 ) 4 2X = 0 1 ( x 2 +1 ) 4 2xdx MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacaGGOaGaamiwamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaigdacaGGPaWaaWbaaSqabeaacaaI0aaaaOGaeyyXICTaaGOmaiaadIfaaSqaaiaaicdaaeaacaaIXaaaniabgUIiYdGccqGH9aqpdaWdXbqaaiaacIcacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGymaiaacMcadaahaaWcbeqaaiaaisdaaaGccqGHflY1caaIYaGaamiEaiaaykW7caWGKbGaamiEaaWcbaGaaGimaaqaaiaaigdaa0Gaey4kIipaaaa@55A5@   by substituting

g= X 2 +1, g =2X MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiabg2da9iaadIfadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIXaGaaiilaiaaywW7ceWGNbGbauaacqGH9aqpcaaIYaGaamiwaaaa@4120@

0 1 ( X 2 +1 ) 4 2X = 0 1 X 4 ( X 2 +1)( X 2 +1 ) = 1 2 X 4 = 1 5 X 5 | 1 2 = 31 5 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@686E@

The first step in the next example is to care for the missing factor 3. This is easily done by considering 1= 1 3 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabg2da9maalaaabaGaaGymaaqaaiaaiodaaaGaeyyXICTaaG4maaaa@3C3C@ , a common trick. Unsuitable factors are no obstacles at all as they are always placeable outside the integral sign.

Example:  To calculate the integral  0 2 X 2 2 X 3 +1 = 0 2 x 2 2 x 3 +1 dx MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaadaWcaaqaaiaadIfadaahaaWcbeqaaiaaikdaaaaakeaacaaIYaWaaOaaaeaacaWGybWaaWbaaSqabeaacaaIZaaaaOGaey4kaSIaaGymaaWcbeaaaaaabaGaaGimaaqaaiaaikdaa0Gaey4kIipakiabg2da9maapehabaWaaSaaaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaaGcbaGaaGOmamaakaaabaGaamiEamaaCaaaleqabaGaaG4maaaakiabgUcaRiaaigdaaSqabaaaaOGaaGPaVlaadsgacaWG4baaleaacaaIWaaabaGaaGOmaaqdcqGHRiI8aaaa@4EB4@   we substitute

g= X 3 +1, g =3 X 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiabg2da9iaadIfadaahaaWcbeqaaiaaiodaaaGccqGHRaWkcaaIXaGaaiilaiaaywW7ceWGNbGbauaacqGH9aqpcaaIZaGaamiwamaaCaaaleqabaGaaGOmaaaaaaa@420B@

0 2 X 2 2 X 3 +1 = 1 3 0 2 3 X 2 2 X 3 +1 = 1 3 0 2 1 2 X ( X 3 +1)( X 3 +1 ) = 1 3 1 9 1 2 X = 1 3 X | 1 9 = 2 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@753A@

In our third example we apply the substitution formula in the right to left direction. Now that the integrand is not of the (fg) g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAgacqWIyiYBcaWGNbGaaiykaiabgwSixlqadEgagaqbaaaa@3D98@ type there is no evident substitution g to be read off. Without experiences some substitutions seem to be quite random and strange.

In this example we choose sine for substitution as the integrand's design is related to the Pythagorean theorem 1 sin 2 = cos 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgkHiTiGacohacaGGPbGaaiOBamaaCaaaleqabaGaaGOmaaaakiabg2da9iGacogacaGGVbGaai4CamaaCaaaleqabaGaaGOmaaaaaaa@4021@ which might be promising. Also, in a previous example we already proved 1 2 (sincos+X) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaaGOmaaaacaGGOaGaci4CaiaacMgacaGGUbGaeyyXICTaci4yaiaac+gacaGGZbGaey4kaSIaamiwaiaacMcaaaa@4280@ to be a primitive of cos 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbWaaWbaaSqabeaacaaIYaaaaaaa@39A8@ .

Example:  We solve the integral  1 1 1 X 2 = 1 1 1 x 2 dx MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaadaGcaaqaaiaaigdacqGHsislcaWGybWaaWbaaSqabeaacaaIYaaaaaqabaaabaGaeyOeI0IaaGymaaqaaiaaigdaa0Gaey4kIipakiabg2da9maapehabaWaaOaaaeaacaaIXaGaeyOeI0IaamiEamaaCaaaleqabaGaaGOmaaaaaeqaaOGaaGPaVlaadsgacaWG4baaleaacqGHsislcaaIXaaabaGaaGymaaqdcqGHRiI8aaaa@4B20@   by substituting

g=sin, g =cos MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiabg2da9iGacohacaGGPbGaaiOBaiaacYcacaaMf8Uabm4zayaafaGaeyypa0Jaci4yaiaac+gacaGGZbaaaa@41C5@

1 1 1 X 2 = π 2 π 2 1 X 2 sinsin = π 2 π 2 1 sin 2 cos = π 2 π 2 cos 2 cos = π 2 π 2 |cos|cos = π 2 π 2 cos 2 = 1 2 (sincos+X) | π 2 π 2 = π 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@B4A2@

In a final example we use both directions of the substitution formula to calculate a primitive function of

1 X 2 r 2 :[r,r] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaaIXaGaeyOeI0YaaSaaaeaacaWGybWaaWbaaSqabeaacaaIYaaaaaGcbaGaamOCamaaCaaaleqabaGaaGOmaaaaaaaabeaakiaacQdacaGGBbGaeyOeI0IaamOCaiaacYcacaWGYbGaaiyxaiabgkziUkabl2riHcaa@44D4@

for an arbitrary r>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabg6da+iaaicdaaaa@38A5@ . Again we take sine for substitution as the new function is quite similar to the last one. This time however the boundaries are variable so that g needs to be reversible. sine itself is not bijective but the restriction sin|[ π 2 , π 2 ] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacMgacaGGUbGaaiiFaiaacUfacqGHsisldaWcaaqaaiabec8aWbqaaiaaikdaaaGaaiilamaalaaabaGaeqiWdahabaGaaGOmaaaacaGGDbaaaa@4233@ is. Its invers function

arcsin= (sin|[ π 2 , π 2 ]) 1 :[1,1][ π 2 , π 2 ] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyyaiaackhacaGGJbGaai4CaiaacMgacaGGUbGaeyypa0JaaiikaiGacohacaGGPbGaaiOBaiaacYhacaGGBbGaeyOeI0YaaSaaaeaacqaHapaCaeaacaaIYaaaaiaacYcadaWcaaqaaiabec8aWbqaaiaaikdaaaGaaiyxaiaacMcadaahaaWcbeqaaiabgkHiTiaaigdaaaGccaGG6aGaai4waiabgkHiTiaaigdacaGGSaGaaGymaiaac2facqGHsgIRcaGGBbGaeyOeI0YaaSaaaeaacqaHapaCaeaacaaIYaaaaiaacYcadaWcaaqaaiabec8aWbqaaiaaikdaaaGaaiyxaaaa@5BF8@ .  

 i

is called inverse sine. We only use the dx-notation for this example.

Example:  We recall that 1 2 (sincos+X) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaaGOmaaaacaGGOaGaci4CaiaacMgacaGGUbGaeyyXICTaci4yaiaac+gacaGGZbGaey4kaSIaamiwaiaacMcaaaa@4280@ is a primitive of cos 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbWaaWbaaSqabeaacaaIYaaaaaaa@39A8@ . As cosine is positive on the range arcsin([1,1])=[ π 2 , π 2 ] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyyaiaackhacaGGJbGaai4CaiaacMgacaGGUbGaaiikaiaacUfacqGHsislcaaIXaGaaiilaiaaigdacaGGDbGaaiykaiabg2da9iaacUfacqGHsisldaWcaaqaaiabec8aWbqaaiaaikdaaaGaaiilamaalaaabaGaeqiWdahabaGaaGOmaaaacaGGDbaaaa@4B27@ we calculate for an arbitrary x[r,r] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaacUfacqGHsislcaWGYbGaaiilaiaadkhacaGGDbaaaa@3DB8@ (using Pythagoras' theorem again)

r x 1 u 2 r 2 du =r r x 1 u 2 r 2 1 r du =r 1 x r 1 t 2 dt         substitute  t= u r ,dt= 1 r du =r arcsin(1) arcsin x r 1 sin 2 z coszdz         substitute  t=sinz,dt=coszdz =r arcsin(1) arcsin x r cos 2 zdz = r 2 (sinzcosz+z) | π 2 arcsin x r = r 2 ( x r cos(arcsin x r )+arcsin x r + π 2 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@1D2E@

and thus have established a primitive function of 1 X 2 r 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaaIXaGaeyOeI0YaaSaaaeaacaWGybWaaWbaaSqabeaacaaIYaaaaaGcbaGaamOCamaaCaaaleqabaGaaGOmaaaaaaaabeaaaaa@3B64@ , namely:

X 2 cos(arcsin X r )+ r 2 arcsin X r = X 2 1 X 2 r 2 + r 2 arcsin X r MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@68DE@

We close this part by proving that integrals are translation-resistant which is easily done with the substitution formula..

Proposition:  Let f be integrable on I and a,bI MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacYcacaWGIbGaeyicI4Saamysaaaa@3ABB@ . For each c MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiabgIGiolabl2riHcaa@39C8@ we have:

a b f = a+c b+c f(Xc) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacaWGMbaaleaacaWGHbaabaGaamOyaaqdcqGHRiI8aOGaeyypa0Zaa8qCaeaacaWGMbGaeSigI8MaaiikaiaadIfacqGHsislcaWGJbGaaiykaaWcbaGaamyyaiabgUcaRiaadogaaeaacaWGIbGaey4kaSIaam4yaaqdcqGHRiI8aaaa@4A17@
[8.3.6]

Proof:  As (Xc ) =1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadIfacqGHsislcaWGJbGabiykayaafaGaeyypa0JaaGymaaaa@3BC4@ we may employ the substitution formula and get

a+c b+c f(Xc) = Xc(a+c) Xc(b+c) f = a b f MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacaWGMbGaeSigI8MaaiikaiaadIfacqGHsislcaWGJbGaaiykaaWcbaGaamyyaiabgUcaRiaadogaaeaacaWGIbGaey4kaSIaam4yaaqdcqGHRiI8aOGaeyypa0Zaa8qCaeaacaWGMbaaleaacaWGybGaeyOeI0Iaam4yaiaacIcacaWGHbGaey4kaSIaam4yaiaacMcaaeaacaWGybGaeyOeI0Iaam4yaiaacIcacaWGIbGaey4kaSIaam4yaiaacMcaa0Gaey4kIipakiabg2da9maapehabaGaamOzaaWcbaGaamyyaaqaaiaadkgaa0Gaey4kIipaaaa@5BF2@

8.2. 8.4.