Primitive of Reciprocal of Sine of a x minus Cosine of a x

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Theorem

$\displaystyle \int \frac {\mathrm d x} {\sin a x - \cos a x} = \frac 1 {a \sqrt 2} \ln \left\vert{\tan \left({\frac {a x} 2 - \frac \pi 8}\right)}\right\vert + C$


Proof

\(\displaystyle \int \frac {\mathrm d x} {\sin a x - \cos a x}\) \(=\) \(\displaystyle \int \frac {\mathrm d x} {\sqrt 2 \cos \left({a x - \dfrac {3 \pi} 4}\right)}\) Sine of x minus Cosine of x: Cosine Form
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {\sqrt 2} \int \sec \left({a x - \dfrac {3 \pi} 4}\right) \ \mathrm d x\) Secant is Reciprocal of Cosine


Let:

\(\displaystyle z\) \(=\) \(\displaystyle a x - \dfrac {3 \pi} 4\)
\(\displaystyle \implies \ \ \) \(\displaystyle \frac {\mathrm d z} {\mathrm d x}\) \(=\) \(\displaystyle a\) Derivative of Identity Function
and Derivatives of Function of $a x + b$
\(\displaystyle \implies \ \ \) \(\displaystyle \frac 1 {\sqrt 2} \int \sec \left({a x - \dfrac {3 \pi} 4}\right) \ \mathrm d x\) \(=\) \(\displaystyle \frac 1 {a \sqrt 2} \int \sec z \ \mathrm d z\) Integration by Substitution
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {a \sqrt 2} \ln \left\vert{\tan \left({\frac z 2 + \frac \pi 4}\right)}\right\vert + C\) Primitive of $\sec a x$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {a \sqrt 2} \ln \left\vert{\tan \left({\frac 1 2 \left({a x - \dfrac {3 \pi} 4}\right) + \frac \pi 4}\right)}\right\vert + C\) substituting for $z$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {a \sqrt 2} \ln \left\vert{\tan \left({\frac {a x} 2 - \frac \pi 8}\right)}\right\vert + C\) simplifying

$\blacksquare$


Also see


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