Browse Questions

# Integrate the function$\int f'(ax+b)[f(ax+b)]^n$

Toolbox:
• If an integral function $f(x)=t,$ then $f'(x)dx=dt$ then $\int f(x)dx=\int tdt$
• (ii)$\int x^n dx=\frac{x^{n+1}}{n+1}+c$
Given $I=\int f'(ax+b)[f(ax+b)]^n$

Let $f(ax+b)=t$

On differentiating with respect to x

$f'(ax+b)dx=dt \qquad =>f'(ax+b)dx=\frac{dt}{a}$

Substituting for t and dt

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

On integrating we get

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

Substituting for t we get

$I=\frac{1}{a}[f(ax+b)]^{n+1} \times \frac{1}{n+1}+c$

$=\frac{1}{a(n+1)}[f(ax+b)]^{n+1}+c$