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

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Theorem

$\displaystyle \int \frac {\d x} {\sin^2 a x \cos^2 a x} = \frac {-2 \cot 2 a x} a + C$


Proof

\(\displaystyle \int \frac {\d x} {\sin^2 a x \cos^2 a x}\) \(=\) \(\displaystyle \int \frac {\d x} {\left({\sin a x \cos a x}\right)^2}\)
\(\displaystyle \) \(=\) \(\displaystyle \int \frac {\d x} {\left({\frac {\sin 2 a x} 2}\right)^2}\) Double Angle Formula for Sine
\(\displaystyle \) \(=\) \(\displaystyle 4 \int \frac {\d x} {\sin^2 2 a x}\) Primitive of Constant Multiple of Function
\(\displaystyle \) \(=\) \(\displaystyle 4 \int \csc^2 2 a x \rd x\) Cosecant is Reciprocal of Sine
\(\displaystyle \) \(=\) \(\displaystyle 4 \frac {-\cot 2 a x} {2 a} + C\) Primitive of Square of Cosecant of a x
\(\displaystyle \) \(=\) \(\displaystyle \frac {-2 \cot 2 a x} a + C\) simplifying

$\blacksquare$


Sources