Question

Evaluate:
$\int {\sin }^{3}x{\cos }^{3}xdx$

Answer 1

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

we know 2 sin x cos x = sin 2x

$\Rightarrow \int (sinxcosx{)}^{3}dx=\frac{1}{8}(2sinxcosx{)}^{3}dx$

$\Rightarrow \frac{1}{8}\int si{n}^{3}2xdx$

we know $si{n}^3\theta =3sin\theta -4si{n}^3\theta$

$si{n}^3\theta =\frac{3sin\theta -si{n}^3\theta }{4}$

$\Rightarrow \frac{1}{8}\times \frac{1}{4}\int 3sin2x-sin6xdx$

$\Rightarrow \frac{1}{32}[\frac{-3cos2x}{2}+\frac{cos6x}{6}]$

$\Rightarrow \frac{-1}{64}[3cos2x-\frac{cos6x}{3}]$