Primitive of Cosine of a x over Sine of a x plus Cosine of a x
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Theorem
- $\ds \int \frac {\cos a x \rd x} {\sin a x + \cos a x} = \frac x 2 + \frac 1 {2 a} \ln \size {\sin a x + \cos a x} + C$
Proof
\(\ds \int \frac {\cos a x \rd x} {\sin a x + \cos a x}\) | \(=\) | \(\ds \int \frac {\paren {\sin a x + \cos a x - \sin a x} \rd x} {\sin a x + \cos a x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\paren {\sin a x + \cos a x} \rd x} {\sin a x + \cos a x} - \int \frac {\sin a x \rd x} {\sin a x + \cos a x}\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \rd x - \int \frac {\sin a x \rd x} {\sin a x + \cos a x}\) | simplification | |||||||||||
\(\ds \) | \(=\) | \(\ds x - \int \frac {\sin a x \rd x} {\sin a x + \cos a x} + C\) | Primitive of Constant | |||||||||||
\(\ds \) | \(=\) | \(\ds x - \paren {\frac x 2 - \frac 1 {2 a} \ln \size {\sin a x + \cos a x} } + C\) | Primitive of $\dfrac {\sin a x} {\sin a x + \cos a x}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac x 2 + \frac 1 {2 a} \ln \size {\sin a x + \cos a x} + C\) | simplifying |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.414$