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
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.405$