Primitive of Square of Cosine of a x over Sine of a x
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Theorem
- $\ds \int \frac {\cos^2 a x \rd x} {\sin a x} = \frac {\cos a x} a + \frac 1 a \ln \size {\tan \frac {a x} 2} + C$
Proof
| \(\ds \int \frac {\cos^2 a x \rd x} {\sin a x}\) | \(=\) | \(\ds \int \frac {\paren {1 - \sin^2 a x} \rd x} {\sin a x}\) | Sum of Squares of Sine and Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \frac {\d x} {\sin a x} - \int \sin a x \rd x\) | Linear Combination of Primitives | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \csc a x \rd x - \int \sin a x \rd x\) | Cosecant is Reciprocal of Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\cos a x} a + \int \csc a x \rd x + C\) | Primitive of $\sin a x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\cos a x} a + \frac 1 a \ln \size {\tan \frac {a x} 2} + C\) | Primitive of $\csc a x$ |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.409$