Primitive of Sine of a x over Sine of a x minus Cosine of a x
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Theorem
- $\ds \int \frac {\sin a x \rd x} {\sin a x - \cos a x} = \frac x 2 + \frac 1 {2 a} \ln \size {\sin a x - \cos a x} + C$
Proof
First note that:
\(\text {(1)}: \quad\) | \(\ds \map {\frac {\d} {\d x} } {\sin a x - \cos a x}\) | \(=\) | \(\ds a \paren {\cos a x + \sin a x}\) | Derivative of $\sin a x$ and Derivative of $\cos a x$ |
Then:
\(\ds \int \frac {\sin a x \rd x} {\sin a x - \cos a x}\) | \(=\) | \(\ds \int \frac {\paren {\sin a x - \cos a x + \cos a x} \rd x} {\sin a x - \cos a x}\) | |||||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\paren {\sin a x - \cos a x} \rd x} {\sin a x - \cos a x} + \int \frac {\cos a x \rd x} {\sin a x - \cos a x}\) | Linear Combination of Primitives | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \rd x + \int \frac {\cos a x \rd x} {\sin a x - \cos a x}\) | simplification | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \rd x + \int \frac {\paren {\cos a x + \sin a x - \sin a x} \rd x} {\sin a x - \cos a x}\) | |||||||||||||
\(\ds \) | \(=\) | \(\ds \int \rd x + \int \frac {\paren {\cos a x + \sin a x} \rd x} {\sin a x - \cos a x} - \int \frac {\sin a x \rd x} {\sin a x - \cos a x}\) | Linear Combination of Primitives | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 \int \frac {\sin a x \rd x} {\sin a x - \cos a x}\) | \(=\) | \(\ds \int \rd x + \int \frac {\paren {\cos a x + \sin a x} \rd x} {\sin a x - \cos a x}\) | rearranging | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \frac {\sin a x \rd x} {\sin a x + \cos a x}\) | \(=\) | \(\ds \frac 1 2 \int \rd x + \frac 1 {2 a} \int \frac {a \paren {\cos a x + \sin a x} \rd x} {\sin a x - \cos a x}\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac x 2 + \frac 1 {2 a} \int \frac {a \paren {\cos a x + \sin a x} \rd x} {\sin a x - \cos a x} + C\) | Primitive of Constant | ||||||||||||
Then from $(1)$: | |||||||||||||||
\(\ds \) | \(=\) | \(\ds \frac x 2 + \frac 1 {2 a} \ln \size {\sin a x - \cos a x} + C\) | Primitive of Function under its Derivative |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.413$