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

Theorem

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

Proof

First note that:

\(\text {(1)}: \quad\)

\(\ds \map {\frac \d {\d x} } {\map \sin {a x + \phi} }\)

\(=\)

\(\ds a \map \cos {a x + \phi}\)

Derivative of $\sin a x$ etc.

Then:

\(\ds \int \frac {\cos a x \rd x} {\map \sin {a x + \phi} }\)

\(=\)

\(\ds \int \frac {\cos a x \cos \phi \rd x} {\cos \phi \map \sin {a x + \phi} }\)

multiplying top and bottom by $\cos \phi$

\(\ds \)

\(=\)

\(\ds \int \frac {\paren {\cos a x \cos \phi - \sin a x \sin \phi + \sin a x \sin \phi} \rd x} {\cos \phi \map \sin {a x + \phi} }\)

\(\ds \)

\(=\)

\(\ds \int \frac {\paren {\map \cos {a x + \phi} + \sin a x \sin \phi} \rd x} {\cos \phi \map \sin {a x + \phi} }\)

Cosine of Sum

\(\ds \)

\(=\)

\(\ds \frac 1 {\cos \phi} \int \frac {\map \cos {a x + \phi} \rd x} {\map \sin {a x + \phi} } + \frac {\sin \phi} {\cos \phi} \int \frac {\sin a x \rd x} {\map \sin {a x + \phi} }\)

Linear Combination of Primitives

\(\ds \)

\(=\)

\(\ds \frac 1 {\cos \phi} \ln \size {\map \sin {a x + \phi} } + \frac {\sin \phi} {\cos \phi} \int \frac {\sin a x \rd x} {\map \sin {a x + \phi} }\)

Primitive of Function under its Derivative and $(1)$

\(\ds \)

\(=\)

\(\ds \frac {\ln \size {\map \sin {a x + \phi} } } {a \cos \phi} + \tan \phi \int \frac {\sin a x \rd x} {\map \sin {a x + \phi} } + C\)

Tangent is Sine divided by Cosine

$\blacksquare$