Primitive of Cosine of a x over Sine of a x minus Cosine of a x

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Theorem

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


Proof

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

$\blacksquare$


Also see


Sources

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