Question

$\int f^{\prime }(ax+b)\left[f\right(ax+b){]}^{n}dx.$

Answer 1

$LetI=\int f^{\prime }(ax+b)\left[f\right(ax+b\left){]}^{n}dxPutf\right(ax+b)=t\Rightarrow f^{\prime }(ax+b)adx=dt=\frac{dt}{a{f}^{\prime }(ax+b)}\therefore I=\int f^{\prime }(ax+b)t^n\frac{dt}{f^{\prime }(ax+b)a}=\frac{1}{a}\int t^{n}dt=\frac{1}{a}\left(\frac{t^{n+1}}{n+1}\right)+C=\frac{1}{a}.\frac{\left[f\right(ax+b){]}^{n+1}}{n+1}+C$