Integration of trigonometric function

[Hashiramasenju]

Homework Statement

I have included the LaTex version of the problem.
$\int \frac{sin^2 x}{1+cos^2 x} dx$

Homework Equations

Simplifying fraction
Partial fractions

The Attempt at a Solution

I have uploaded my attempt at the solution.

 

[LCKurtz]

I don't usually read handwritten solutions, and I never read ones printed sideways. Regarding your integral, I would suggest suggest using the double angle cosine formulas for your squared trig functions. This will give you a rational function of $\cos(2x)$. Then the tangent half angle substitution will help you out. See:

https://en.wikipedia.org/wiki/Tangent_half-angle_substitution

 

[SteamKing]

Partial fractions and trig functions usually don't go together. Partial fractions are typically used to simplify rational expressions of a single variable.

For integrating rational expressions of trig functions, a typical approach is to simplify using a trig identity of some sort first, and then try to use either u-substitution or integration by parts, if those techniques might be useful.