Primitive of Sine of a x over Sine of a x plus phi

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Theorem

$\ds \int \frac {\sin a x \rd x} {\map \sin {a x + \phi} } = \frac x {\cos \phi} - \tan \phi \int \frac {\cos a x \rd x} {\map \sin {a x + \phi} } + C$


Proof

\(\ds \int \frac {\sin a x \rd x} {\map \sin {a x + \phi} }\) \(=\) \(\ds \frac 1 {\cos \phi} \int \frac {\sin a x \cos \phi \rd x} {\map \sin {a x + \phi} }\) multiplying top and bottom by $\cos \phi$
\(\ds \) \(=\) \(\ds \frac 1 {\cos \phi} \int \frac {\paren {\sin a x \cos \phi + \cos a x \sin \phi - \cos a x \sin \phi} \rd x} {\map \sin {a x + \phi} }\)
\(\ds \) \(=\) \(\ds \frac 1 {\cos \phi} \int \frac {\paren {\map \sin {a x + \phi} - \cos a x \sin \phi} \rd x} {\map \sin {a x + \phi} }\) Sine of Sum
\(\ds \) \(=\) \(\ds \frac 1 {\cos \phi} \int \frac {\paren {\map \sin {a x + \phi} - \cos a x \sin \phi} \rd x} {\map \sin {a x + \phi} }\) Sine of Sum
\(\ds \) \(=\) \(\ds \frac 1 {\cos \phi} \int \d x - \frac {\sin \phi} {\cos \phi} \int \frac {\cos a x \rd x} {\map \sin {a x + \phi} }\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \frac x {\cos \phi} - \tan \phi \int \frac {\cos a x \rd x} {\map \sin {a x + \phi} } + C\) Tangent is Sine divided by Cosine

$\blacksquare$