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Integrate the function
$\int[f(a x+b)]^n f^{\prime}(a x+b) d x$
$\int[f(a x+b)]^n f^{\prime}(a x+b) d x$
Solution:
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Put $f(a x+b)=t$ so that $f^{\prime}(a x+b) \cdot a d x=d t$
$\therefore \mathrm{I}=\frac{1}{\mathrm{a}} \int \mathrm{t}^{\mathrm{n}} \mathrm{dt}=\frac{1}{\mathrm{a}} \frac{\mathrm{t}^{\mathrm{n}+1}}{\mathrm{n}+1}+\mathrm{c}=\frac{[\mathrm{f}(\mathrm{ax}+\mathrm{b})]^{\mathrm{n}+1}}{\mathrm{a}(\mathrm{n}+1)}+\mathrm{c}$
$\therefore \mathrm{I}=\frac{1}{\mathrm{a}} \int \mathrm{t}^{\mathrm{n}} \mathrm{dt}=\frac{1}{\mathrm{a}} \frac{\mathrm{t}^{\mathrm{n}+1}}{\mathrm{n}+1}+\mathrm{c}=\frac{[\mathrm{f}(\mathrm{ax}+\mathrm{b})]^{\mathrm{n}+1}}{\mathrm{a}(\mathrm{n}+1)}+\mathrm{c}$
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