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 |
$\blacksquare$