Primitive of Function of Sine and Cosine

Theorem

 * $\displaystyle \int F \left({\sin x, \cos x}\right)\ \mathrm d x = 2 \int F \left({\frac {2 u} {1 + u^2}, \frac {1 - u^2} {1 + u^2} }\right) \frac {\mathrm d u} {1 + u^2}$

where $u = \tan \dfrac x 2$.

Proof
Let $u = \tan \dfrac x 2$. Then $x = 2 \tan^{-1} u$

Then:

The result follows from Integration by Substitution.