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

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Theorem

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


Proof

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


Let:

\(\ds z\) \(=\) \(\ds a x - \dfrac {3 \pi} 4\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d z} {\d x}\) \(=\) \(\ds a\) Derivative of Identity Function
and Derivatives of Function of $a x + b$
\(\ds \leadsto \ \ \) \(\ds \frac 1 {\sqrt 2} \int \map \sec {a x - \dfrac {3 \pi} 4} \rd x\) \(=\) \(\ds \frac 1 {a \sqrt 2} \int \sec z \rd z\) Integration by Substitution
\(\ds \) \(=\) \(\ds \frac 1 {a \sqrt 2} \ln \size {\map \tan {\frac z 2 + \frac \pi 4} } + C\) Primitive of $\sec a x$
\(\ds \) \(=\) \(\ds \frac 1 {a \sqrt 2} \ln \size {\map \tan {\frac 1 2 \paren {a x - \dfrac {3 \pi} 4} + \frac \pi 4} } + C\) substituting for $z$
\(\ds \) \(=\) \(\ds \frac 1 {a \sqrt 2} \ln \size {\map \tan {\frac {a x} 2 - \frac \pi 8} } + C\) simplifying

$\blacksquare$


Also see


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