Question

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

Answer 1

Let

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

Substituting $f(ax+b)=t$

Differentiating w.r.t. $x$

$f^{\prime }(ax+b).a=\frac{dt}{dx}$

$\Rightarrow f^{\prime }(ax+b)dx=\frac{dt}{a}$

Now,

$I=\int t^n\frac{dt}{a}=\frac{1}{a}\int t^{n}dt$

$\Rightarrow I=\frac{1}{a}\left[\frac{t^{n+1}}{n+1}\right]+C$

$\Rightarrow I=\frac{{\left[f\right(ax+b\left)\right]}^{n+1}}{a(n+1)}+C$

Where $C$ is constant of integration