Question

How do you find the integral of `cos(x)/(5+sin^2(x))dx`?

Answer 1

Given: `int cos(x)/(5+sin^2(x))dx`

Let `u = sin(x)`, then `du = cos(x)dx)`:

`int cos(x)/(5+sin^2(x))dx = int 1/(5+u^2)du`

You should recognize the right side as the inverse tangent form:

`int cos(x)/(5+sin^2(x))dx = sqrt5/5tan^-1(u/sqrt5)+ C`

Reverse the substitution for u:

`int cos(x)/(5+sin^2(x))dx = sqrt5/5tan^-1(sin(x)/sqrt5)+ C`