Primitive of Square of Cosine of a x over Sine of a x

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Theorem

$\displaystyle \int \frac {\cos^2 a x \ \mathrm d x} {\sin a x} = \frac {\cos a x} a + \frac 1 a \ln \left\vert{\tan \frac {a x} 2}\right\vert + C$


Proof

\(\displaystyle \int \frac {\cos^2 a x \ \mathrm d x} {\sin a x}\) \(=\) \(\displaystyle \int \frac {\left({1 - \sin^2 a x}\right) \ \mathrm d x} {\sin a x}\) Sum of Squares of Sine and Cosine
\(\displaystyle \) \(=\) \(\displaystyle \int \frac {\mathrm d x} {\sin a x} - \int \sin a x \ \mathrm d x\) Linear Combination of Integrals
\(\displaystyle \) \(=\) \(\displaystyle \int \csc a x \ \mathrm d x - \int \sin a x \ \mathrm d x\) Cosecant is Reciprocal of Sine
\(\displaystyle \) \(=\) \(\displaystyle \frac {\cos a x} a + \int \csc a x \ \mathrm d x + C\) Primitive of $\sin a x$
\(\displaystyle \) \(=\) \(\displaystyle \frac {\cos a x} a + \frac 1 a \ln \left\vert{\tan \frac {a x} 2}\right\vert + C\) Primitive of $\csc a x$

$\blacksquare$


Also see


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