Primitive of Reciprocal of Sine of a x by Cosine of a x
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Theorem
- $\ds \int \frac {\d x} {\sin a x \cos a x} = \frac 1 a \ln \size {\tan a x} + C$
Proof
| \(\ds \int \frac {\d x} {\sin a x \cos a x}\) | \(=\) | \(\ds \int \frac {\sec a x \rd x} {\sin a x}\) | Secant is Reciprocal of Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \frac {\sec^2 a x \rd x} {\sin a x \sec a x}\) | multiplying top and bottom by $\sec a x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \frac {\sec^2 a x \rd x} {\frac {\sin a x} {\cos a x} }\) | Secant is Reciprocal of Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \frac {\sec^2 a x \rd x} {\tan a x}\) | Tangent is Sine divided by Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 a \ln \size {\tan a x} + C\) | Primitive of $\dfrac {\sec^2 a x} {\tan a x}$ |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.404$