# Primitive of Reciprocal of Sine of a x plus Cosine of a x/Proof 1

## Theorem

$\displaystyle \int \frac {\d x} {\sin a x + \cos a x} = \frac 1 {a \sqrt 2} \ln \size {\map \tan {\frac {a x} 2 + \frac \pi 8} } + C$

## Proof

 $\displaystyle \int \frac {\d x} {\sin a x + \cos a x}$ $=$ $\displaystyle \int \frac {\d x} {\sqrt 2 \map \cos {a x - \dfrac \pi 4} }$ Sine of x plus Cosine of x: Cosine Form $\displaystyle$ $=$ $\displaystyle \frac 1 {\sqrt 2} \int \map \sec {a x - \dfrac \pi 4} \rd x$ Secant is Reciprocal of Cosine

Let:

 $\displaystyle z$ $=$ $\displaystyle a x - \dfrac \pi 4$ $\displaystyle \leadsto \ \$ $\displaystyle \frac {\d z} {\d x}$ $=$ $\displaystyle a$ Derivative of Identity Function and Derivatives of Function of $a x + b$ $\displaystyle \leadsto \ \$ $\displaystyle \frac 1 {\sqrt 2} \int \map \sec {a x - \dfrac \pi 4} \rd x$ $=$ $\displaystyle \frac 1 {a \sqrt 2} \int \sec z \rd z$ Integration by Substitution $\displaystyle$ $=$ $\displaystyle \frac 1 {a \sqrt 2} \ln \size {\map \tan {\frac z 2 + \frac \pi 4} } + C$ Primitive of $\sec a x$ $\displaystyle$ $=$ $\displaystyle \frac 1 {a \sqrt 2} \ln \size {\map \tan {\frac 1 2 \paren {a x - \dfrac \pi 4} + \frac \pi 4} } + C$ substituting for $z$ $\displaystyle$ $=$ $\displaystyle \frac 1 {a \sqrt 2} \ln \size {\map \tan {\frac {a x} 2 + \frac \pi 8} } + C$ simplifying

$\blacksquare$