Primitive of Cosine of a x over Sine of a x plus phi
Jump to navigation
Jump to search
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 |
$\blacksquare$