Primitive of Reciprocal of Sine of a x by Square of Cosine of a x

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Theorem

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


Proof

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

$\blacksquare$


Also see


Sources