Question

$\int co{s}^{4}xdx$

Answer 1

Consider given integration,

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

$\Rightarrow \int {\left({\cos }^{2}x\right)}^{2}dx$

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

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

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

$\Rightarrow \int \frac{(1+\frac{1+\cos 2.2x}{2}+2\cos 2x)}{4}dx$

$\Rightarrow \int \frac{(2+\cos 4x+4\cos 2x)}{8}dx$

$\Rightarrow \frac{1}{8}\int (2+\cos 4x+4\cos 2x)dx$

$\Rightarrow \frac{1}{8}(2x+\frac{\sin 4x}{4}+4\frac{\sin 2x}{2})+C$

$\Rightarrow \frac{1}{8}(2x+\frac{\sin 4x}{4}+\sin 2x)+C$

Hence, this is the answer.