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

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Theorem

$\ds \int \frac {\d x} {\sin a x \cos^2 a x} = \frac 1 a \ln \size {\tan \frac {a x} 2} + \frac 1 {a \cos a x} + C$


Proof

\(\ds \int \frac {\d x} {\sin a x \cos^2 a x}\) \(=\) \(\ds \int \frac {\paren {\sin^2 a x + \cos^2 a x} \rd x} {\sin a x \cos^2 a x}\) Sum of Squares of Sine and Cosine
\(\ds \) \(=\) \(\ds \int \frac {\sin^2 a x \rd x} {\sin a x \cos^2 a x} + \int \frac {\cos^2 a x \rd x} {\sin a x \cos^2 a x}\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \int \frac {\d x} {\sin a x} + \int \frac {\sin a x \rd x} {\cos^2 a x}\) simplifying
\(\ds \) \(=\) \(\ds \int \csc a x \rd x + \int \frac {\sin a x \rd x} {\cos^2 a x}\) Cosecant is Reciprocal of Sine
\(\ds \) \(=\) \(\ds \int \csc a x \rd x + \int \frac {\tan a x \rd x} {\cos a x}\) Tangent is Sine divided by Cosine
\(\ds \) \(=\) \(\ds \int \csc a x \rd x + \int \sec a x \tan a x \rd x\) Secant is Reciprocal of Cosine
\(\ds \) \(=\) \(\ds \frac 1 a \ln \size {\tan \frac {a x} 2} + \int \sec a x \tan a x \rd x + C\) Primitive of $\csc a x$
\(\ds \) \(=\) \(\ds \frac 1 a \ln \size {\tan \frac {a x} 2} + \frac 1 {a \cos a x} + C\) Primitive of $\sec^n a x \tan a x$

$\blacksquare$


Also see


Sources

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