Integrate the function in Exercise: f (ax+b ) [f(ax+b) ]^ n - Vidyadip

Question

Integrate the function in Exercise:
$\text{f}\big(\text{ax}+\text{b}\big)\big[\text{f(ax}+\text{b)}\big]^{\text{n}}$

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Answer

$\text{f' (ax}+\text{b)}\big[\text{f (ax}+\text{b)}\big]^{\text{n}}$ $\text{Let}\ \text{f (ax}+\text{b)}=\text{t}\Rightarrow \text{af}\ \text{(ax}+\text{b)}\text{dx}=\text{dt}$ $\Rightarrow\int\text{f}\ \big(\text{ax}+\text{b}\big)\big[\text{f}\text{(ax}+\text{b)}\big]^{\text{n}}\text{dx}=\frac{1}{\text{a}}\int\text{t}^{\text{n}}\text{dt}$$=\frac{1}{\text{a}}\bigg[\frac{\text{t}^{\text{n}+1}}{\text{n}+1}\bigg]$
$=\frac{1}{\text{a}(\text{n}+1)}\big(\text{f (ax}+\text{b)}\big)^{\text{n+1}}+\text{C}$

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