Primitive of Reciprocal of Cube of Sine of a x
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Theorem
- $\ds \int \frac {\d x} {\sin^3 a x} = \frac {-\cos a x} {2 a \sin^2 a x} + \frac 1 {2 a} \ln \size {\tan \frac {a x} 2} + C$
Proof
\(\ds \int \frac {\d x} {\sin^3 a x}\) | \(=\) | \(\ds \int \csc^3 a x \rd x\) | Cosecant is $\dfrac 1 \sin$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac{-\csc a x \cot a x} {2 a} + \frac 1 2 \int \csc a x \rd x\) | Primitive of $\csc^n a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-\cos a x} {2 a \sin^2 a x} + \frac 1 2 \int \csc a x \rd x\) | Cosecant is $\dfrac 1 \sin$ and Cotangent is $\dfrac \cos \sin$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-\cos a x} {2 a \sin^2 a x} + \frac 1 {2 a} \ln \size {\tan \frac {a x} 2} + C\) | 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 $\sin a x$: $14.352$