Question

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

Answer 1

Given, $\int {\cos }^{3}xdx=\int {\cos }^{2}x\cos xdx=\int (1-{\sin }^{2}x)\cos xdx$

Let, $\sin x=u$

$\Rightarrow du=\cos xdx$

$\therefore \int (1-{\sin }^{2}x)\cos xdx=\int (1-u^2)du=\int du-\int u^{2}du=u-\frac{u^3}{3}+C=\sin x-\frac{{\sin }^{3}x}{3}+C$

Hence, $\int {\cos }^{3}xdx=\sin x-\frac{{\sin }^{3}x}{3}+C.$