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

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Theorem

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


Proof

First note that:

\(\text {(1)}: \quad\) \(\displaystyle \map {\frac {\d} {\d x} } {\sin a x - \cos a x}\) \(=\) \(\displaystyle a \paren {\cos a x + \sin a x}\) Derivative of $\sin a x$ and Derivative of $\cos a x$


Then:

\(\displaystyle \int \frac {\sin a x \rd x} {\sin a x - \cos a x}\) \(=\) \(\displaystyle \int \frac {\paren {\sin a x - \cos a x + \cos a x} \rd x} {\sin a x - \cos a x}\)
\(\displaystyle \) \(=\) \(\displaystyle \int \frac {\paren {\sin a x - \cos a x} \rd x} {\sin a x - \cos a x} + \int \frac {\cos a x \rd x} {\sin a x - \cos a x}\) Linear Combination of Integrals
\(\displaystyle \) \(=\) \(\displaystyle \int \rd x + \int \frac {\cos a x \rd x} {\sin a x - \cos a x}\) simplification
\(\displaystyle \) \(=\) \(\displaystyle \int \rd x + \int \frac {\paren {\cos a x + \sin a x - \sin a x} \rd x} {\sin a x - \cos a x}\)
\(\displaystyle \) \(=\) \(\displaystyle \int \rd x + \int \frac {\paren {\cos a x + \sin a x} \rd x} {\sin a x - \cos a x} - \int \frac {\sin a x \rd x} {\sin a x - \cos a x}\) Linear Combination of Integrals
\(\displaystyle \leadsto \ \ \) \(\displaystyle 2 \int \frac {\sin a x \rd x} {\sin a x - \cos a x}\) \(=\) \(\displaystyle \int \rd x + \int \frac {\paren {\cos a x + \sin a x} \rd x} {\sin a x - \cos a x}\) rearranging
\(\displaystyle \leadsto \ \ \) \(\displaystyle \int \frac {\sin a x \rd x} {\sin a x + \cos a x}\) \(=\) \(\displaystyle \frac 1 2 \int \rd x + \frac 1 {2 a} \int \frac {a \paren {\cos a x + \sin a x} \rd x} {\sin a x - \cos a x}\) Linear Combination of Integrals
\(\displaystyle \) \(=\) \(\displaystyle \frac x 2 + \frac 1 {2 a} \int \frac {a \paren {\cos a x + \sin a x} \rd x} {\sin a x - \cos a x} + C\) Primitive of Constant
Then from $(1)$:
\(\displaystyle \) \(=\) \(\displaystyle \frac x 2 + \frac 1 {2 a} \ln \size {\sin a x - \cos a x} + C\) Primitive of Function under its Derivative

$\blacksquare$


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