Question

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

Answer 1

Consider the given integral.

$I=\int {\cos }^{2}xdx$

Since,

$\cos 2x=2{\cos }^{2}x-1$

${\cos }^{2}x=\frac{\cos 2x+1}{2}$

Therefore,

$I=\int \frac{\cos 2x+1}{2}dx$

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

$I=\frac{1}{2}(\frac{\sin 2x}{2}+x)+C$

$I=\frac{\sin 2x}{4}+\frac{x}{2}+C$

Hence, this is the answer.