Primitive of Reciprocal of Sine of a x/Logarithm of Tangent Form
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Theorem
- $\ds \int \frac {\d x} {\sin a x} = \frac 1 a \ln \size {\tan \frac {a x} 2} + C$
Proof
\(\ds \int \frac {\d x} {\sin x}\) | \(=\) | \(\ds \int \csc x \rd x\) | Definition of Real Cosecant Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \ln \size {\tan \frac x 2} + C\) | Primitive of $\csc x$: Tangent Form | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \frac {\d x} {\sin a x}\) | \(=\) | \(\ds \frac 1 a \ln \size {\tan \frac {a x} 2} + C\) | Primitive of Function of Constant Multiple |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$: $14.345$