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Question

Mathematics Question on integral

Integrate the function: ƒ′(ax+b)[ƒ(ax+b)]nƒ'(ax+b)[ƒ(ax+b)]^n

Answer

ƒ′(ax+b)[ƒ(ax+b)]nƒ'(ax+b)[ƒ(ax+b)]^n

Let ƒ(ax+b)=t⇒aƒ′(ax+b)dx=dtLet \ ƒ(ax+b)=t⇒aƒ'(ax+b)dx=dt

⇒ƒ′(ax+b)[ƒ(ax+b)]ndx=1a∫tndt⇒ƒ'(ax+b)[ƒ(ax+b)]^ndx=\frac {1}{a}∫t^ndt

                                                   =$\frac 1a[\frac {t^{n+1}}{n+1}]$

                                                   =$\frac {1}{a(n+1)}(ƒ(ax+b))^{n+1}+C$