Question

Solve:
$\int {\cos }^{4}2xdx$

Answer 1

$\int {\cos }^{4}2xdx$

$=\frac{1}{4}\int {\left(2{\cos }^{2}2x\right)}^{2}dx$

$=\frac{1}{4}\int {(1+\cos 4x)}^{2}dx$

$=\frac{1}{4}\int (1+{\cos }^{2}4x+2\cos 4x)dx$

$=\frac{1}{4}\int dx+\frac{1}{4}\int {\cos }^{2}4xdx+\frac{2}{4}\int \cos 4xdx$

$=\frac{x}{4}+\frac{1}{8}\int (1+\cos 8x)dx+\frac{1}{2}\times \frac{\sin 4x}{4}+c$

$=\frac{x}{4}+\frac{x}{8}+\frac{\sin 8x}{64}+\frac{\sin 4x}{8}+c$

$=\frac{3}{8}+\frac{\sin 8x}{64}+\frac{\sin 4x}{8}+c$