Question

Evaluate

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

Answer 1

We have,

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

Then,

Using formula and we get,

$\int \frac{1}{x^2+1}dx={\tan }^{-1}x$

Then,

$\int \frac{dx}{{\cos }^{2}x+1}={\tan }^{-1}\cos x$

$\int \frac{dx}{{\cos }^{2}x+1}={\tan }^{-1}\tan \frac{\sqrt{1-x^2}}{x}$

$\int \frac{dx}{{\cos }^{2}x+1}=\frac{\sqrt{1-x^2}}{x}$

Hence, this is the answer.