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 $\csc a x$

$\blacksquare$


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