Question

Evaluate:

$\int {\sin }^{3}x.{\cos }^{3}xdx$

Answer 1

$\int si{n}^{3}xco{s}^{3}xdx$

$\Rightarrow \int (1-co{s}^{2}x)sinxco{s}^{3}xdx$

$\int si{n}^{3}xcosx(1-si{n}^{2}x)dx$

Let $sinx=m$

Differentiating w.r.t. $x$, we get,

$cosxdx=dm$

$\Rightarrow \int m^3(1-m^2)dm$

$\Rightarrow \int m^{3}dm-\int m^{5}dm$

$\Rightarrow \frac{m^4}{4}-\frac{m^6}{6}+c$

Back substituting the value of $m$

$\Rightarrow \frac{si{n}^{4}x}{4}-\frac{si{n}^{6}x}{6}+c$