Integrate the following functions w.r.t. x: ^4 x ^3 x - Vidyadip

Question

Integrate the following functions w.r.t. x:
$\sin ^4 x \cdot \cos ^3 x$

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Answer

Let $I=\int \sin ^4 x \cdot \cos ^3 x d x$
$=\int \sin ^4 x \cdot \cos ^2 x \cdot \cos x d x$
$=\int \sin ^4 x\left(1-\sin ^2 x\right) \cos x d x$
Put $\sin x=t \quad \therefore \cos x d x=d t$
$\therefore I =\int t^4\left(1-t^2\right) d t=\int\left(t^4-t^6\right) d t$
$ =\int t^4 d t-\int t^6 d t$
$ =\frac{t^5}{5}-\frac{t^7}{7}+c$
$=\frac{1}{5} \sin ^5 x-\frac{1}{7} \sin ^7 x+c$

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