Primitive of Reciprocal of Square of Sine of a x by Square of Cosine of a x
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Theorem
- $\ds \int \frac {\d x} {\sin^2 a x \cos^2 a x} = \frac {-2 \cot 2 a x} a + C$
Proof
\(\ds \int \frac {\d x} {\sin^2 a x \cos^2 a x}\) | \(=\) | \(\ds \int \frac {\d x} {\left({\sin a x \cos a x}\right)^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\d x} {\left({\frac {\sin 2 a x} 2}\right)^2}\) | Double Angle Formula for Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 \int \frac {\d x} {\sin^2 2 a x}\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 \int \csc^2 2 a x \rd x\) | Cosecant is Reciprocal of Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 \frac {-\cot 2 a x} {2 a} + C\) | Primitive of Square of Cosecant of a x | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-2 \cot 2 a x} a + C\) | simplifying |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.407$