Question

Evaluate the integral `int \ sinx/(2+cos^2x) \ dx`?

Answer 1

`int \ sinx/(2+cos^2x) \ dx = -sqrt(2)/2 arctan(cosx/sqrt(2)) + C`

Explanation:

We seek:

`I = int \ sinx/(2+cos^2x) \ dx`

If we look at the denominator then a substitution:

`2u^2 = cos^2x`

looks promising, so let us try the substitution:

`u = cosx/sqrt(2) => 2u^2 = cos^2x`

Differentiating implicitly wrt `x`:

`(du)/dx = -sinx/sqrt(2)`

So substituting into the integral, it becomes:

`I = int \ (-sqrt(2))/(2+2u^2) \ du`
`\ \ = -sqrt(2) \ int \ 1/(2(1+u^2)) \ du`
`\ \ = -sqrt(2)/2 \ int \ 1/(1+u^2) \ du`

This is now a standard result, thus:

`I = -sqrt(2)/2 arctan(u) + C`

Then restoring the substitution:

`I = -sqrt(2)/2 arctan(cosx/sqrt(2)) + C`