Question

Integrate:

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

Answer 1

$f\left(x\right)=\frac{1}{1-cos\left(2x\right)}$

Taking integration from both sides,we get

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

$\int f\left(x\right)dx=\int \frac{1}{2sin2\left(x\right)dx}$

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

$\int f\left(x\right)dx=-\frac{1}{2}cot\left(x\right)+C$