# Primitive of Cosine of a x over Sine of a x plus Cosine of a x

## Theorem

$\displaystyle \int \frac {\cos a x \rd x} {\sin a x + \cos a x} = \frac x 2 + \frac 1 {2 a} \ln \size {\sin a x + \cos a x} + C$

## Proof

 $\displaystyle \int \frac {\cos a x \rd x} {\sin a x + \cos a x}$ $=$ $\displaystyle \int \frac {\paren {\sin a x + \cos a x - \sin a x} \rd x} {\sin a x + \cos a x}$ $\displaystyle$ $=$ $\displaystyle \int \frac {\paren {\sin a x + \cos a x} \rd x} {\sin a x + \cos a x} - \int \frac {\sin a x \rd x} {\sin a x + \cos a x}$ Linear Combination of Integrals $\displaystyle$ $=$ $\displaystyle \int \rd x - \int \frac {\sin a x \rd x} {\sin a x + \cos a x}$ simplification $\displaystyle$ $=$ $\displaystyle x - \int \frac {\sin a x \rd x} {\sin a x + \cos a x} + C$ Primitive of Constant $\displaystyle$ $=$ $\displaystyle x - \paren {\frac x 2 - \frac 1 {2 a} \ln \size {\sin a x + \cos a x} } + C$ Primitive of $\dfrac {\sin a x} {\sin a x + \cos a x}$ $\displaystyle$ $=$ $\displaystyle \frac x 2 + \frac 1 {2 a} \ln \size {\sin a x + \cos a x} + C$ simplifying

$\blacksquare$