Question

Integrate:

$\int {\cos }^{3}xdx$

Answer 1

Q. Integrate $\int co{s}^{3}xdx$

$[\because$ We know that, for any n]

(1) $\int co{s}^{n}xdx=\frac{cos{x}^{n-1}.sinx}{n}+\frac{(n-1)}{n}{I}_{n-2}$

(2) $\int si{n}^{n}xdx=\frac{-sin{x}^{n-1}.cosx}{n}+\frac{(n-1)}{n}{I}_{n-2}]$

$\therefore \int co{s}^{3}xdx=\frac{cos{x}^{3-1}.sinx}{3}+\frac{(3-1)}{3}{I}_{3-2}$

$=\frac{cos{x}^2.sinx}{3}+\frac{2}{3}{I}_1$

$I_1=\int co{s}^{1}xdx$

$=sinx+C$.

$\therefore \int co{s}^{3}xdx=\frac{cos{x}^2.sinx}{3}+\frac{2}{3}sinx+C$