Primitive of Reciprocal of Square of Sine of a x

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Theorem

$\displaystyle \int \frac {\mathrm d x} {\sin^2 a x} = \frac {-\cot a x} a + C$


Proof

\(\displaystyle \int \frac {\mathrm d x} {\sin^2 a x}\) \(=\) \(\displaystyle \int \csc^2 a x \ \mathrm d x\) Definition of Cosecant
\(\displaystyle \) \(=\) \(\displaystyle \frac {-\cot a x} a + C\) Primitive of $\csc^2 a x$

$\blacksquare$


Also see


Sources