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

$\blacksquare$


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