Question

Find the integrals of the functions.
$\int si{n}^{3}x.co{s}^{3}xdx$

Answer 1

$\int si{n}^{3}x.co{s}^{3}xdx$
Let $cosx=t\Rightarrow -sinx=\frac{dt}{dx}\Rightarrow dx=\frac{dt}{-sinx}$
$\therefore \int si{n}^{3}xco{s}^{3}xdx=\int si{n}^{3}x.t^3\frac{dt}{-sinx}=-\int si{n}^{2}x.t^{3}dt=-\int t^3(1-co{s}^2)dt=-\int t^3.(1-t^2)dt=-\int [t^3-t^5]dt=-[\frac{t^4}{4}-\frac{t^6}{6}]+C=\frac{co{s}^{6}x}{6}-\frac{co{s}^{4}x}{4}+C$