Integrate the function f^ (a x+b)[f(a x+b)]^ n - Vidyadip

Question

Integrate the function $f^{\prime}(a x+b)[f(a x+b)]^{n}$

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Answer

Let f(ax + b) = t $\Rightarrow$ a.f'(ax + b)dx = dt
$\Rightarrow \int \mathrm{f}^{\prime}(\mathrm{ax}+\mathrm{b})\left[\mathrm{f}(\mathrm{ax}+\mathrm{b})^{\mathrm{n}}\right]=\int \mathrm{t}^{\mathrm{n}}\left(\frac{\mathrm{dt}}{\mathrm{a}}\right)$
$=\frac{1}{a} \int t^{n} d t$
= $\frac{1}{a} \cdot \frac{t^{n+1}}{n+1}+c$
= $\frac{1}{a} \cdot \frac{(f(a x+b))^{n+1}}{n+1}+C$
= $\frac{1}{a(n+1)} \cdot(f(a x+b))^{n+1}+c$

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