$\int (3 sin x {cos}^{2} x - {sin}^{3} x) d x$

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Solution

$\int (3 s i n x c o s^{2} x - s i n^{3} x) d x$
$= \int s i n x (3 c o s^{2} x - s i n^{2} x) d x$
$= \int s i n x (3 (1 - s i n^{2} x) - s i n^{2} x) d x$
$= \int s i n x (3 - 4 s i n^{2} x) d x$
$= \int 3 s i n x - 4 s i n^{3} x d x$
$= \int s i n 3 x d x$
$= (- \frac{c o s 3 x}{3}) + c$

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