Question

Integrate the function $f^{\prime }(ax+b)\left[f\right(ax+b){]}^n$

Answer 1

$f^{\prime }(ax+b)\left[f\right(ax+b){]}^n$
Let $f(ax+b)=t\Rightarrow a{f}^{\prime }(ax+b)dx=dt$
$\therefore \int f^{\prime }(ax+b)\left[f\right(ax+b){]}^{n}dx=\frac{1}{a}\int t^{n}dt$
$=\frac{1}{a}\left[\frac{t^{n+1}}{n+1}\right]$
$=\frac{1}{a(n+1)}\left(f\right(ax+b){)}^{n+1}+C$