Primitive of Reciprocal of q plus p by Cosecant of a x
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Theorem
- $\ds \int \frac {\d x} {q + p \csc a x} = \frac x q - \frac p q \int \frac {\d x} {p + q \sin a x} + C$
Proof
\(\ds \int \frac {\d x} {q + p \csc a x}\) | \(=\) | \(\ds \frac 1 q \int \frac {q \rd x} {q + p \csc a x}\) | multiplying top and bottom by $q$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 q \int \frac {\paren {q + p \csc a x - p \csc a x} \rd x} {q + p \csc a x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 q \int \frac {\paren {q + p \csc a x} \rd x} {q + p \csc a x} - \frac p q \int \frac {\csc a x \rd x} {q + p \csc a x}\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 q \int \d x - \frac p q \int \frac {\csc a x \rd x} {q + p \csc a x}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac x q - \frac p q \int \frac {\csc a x \rd x} {q + p \csc a x} + C\) | Primitive of Constant | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac x q - \frac p q \int \frac {\sin a x \, \csc a x \rd x} {q \sin a x + p \sin a x \, \csc a x} + C\) | multiplying top and bottom by $\sin a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac x q - \frac p q \int \frac {\d x} {p + q \sin a x} + C\) | Cosecant is Reciprocal of Sine |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\csc a x$: $14.469$