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

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Theorem

$\ds \int \frac {\d x} {\sin^2 a x \cos a x} = \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } - \frac 1 {a \sin a x} + C$


Proof

\(\ds \int \frac {\d x} {\sin^2 a x \cos a x}\) \(=\) \(\ds \int \frac {\paren {\sin^2 a x + \cos^2 a x} \rd x} {\sin^2 a x \cos a x}\) Sum of Squares of Sine and Cosine
\(\ds \) \(=\) \(\ds \int \frac {\sin^2 a x \rd x} {\sin^2 a x \cos a x} + \int \frac {\cos^2 a x \rd x} {\sin^2 a x \cos a x}\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \int \frac {\d x} {\cos a x} + \int \frac {\cos a x \rd x} {\sin^2 a x}\) simplifying
\(\ds \) \(=\) \(\ds \int \sec a x \rd x + \int \frac {\cos a x \rd x} {\sin^2 a x}\) Secant is Reciprocal of Cosine
\(\ds \) \(=\) \(\ds \int \sec a x \rd x + \int \frac {\cot a x \rd x} {\sin a x}\) Cotangent is Cosine divided by Sine
\(\ds \) \(=\) \(\ds \int \sec a x \rd x + \int \csc a x \cot a x \rd x\) Cosecant is Reciprocal of Sine
\(\ds \) \(=\) \(\ds \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + \int \csc a x \cot a x \rd x + C\) Primitive of $\sec a x$
\(\ds \) \(=\) \(\ds \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } - \frac 1 {a \sin a x} + C\) Primitive of $\csc^n a x \cot a x$

$\blacksquare$


Also see


Sources