Primitive of Reciprocal of q plus p by Secant of a x

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Theorem

$\displaystyle \int \frac {\mathrm d x} {q + p \sec a x} = \frac x q - \frac p q \int \frac {\mathrm d x} {p + q \cos a x} + C$


Proof

\(\displaystyle \int \frac {\mathrm d x} {q + p \sec a x}\) \(=\) \(\displaystyle \frac 1 q \int \frac {q \ \mathrm d x} {q + p \sec a x}\) multiplying top and bottom by $q$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 q \int \frac {\left({q + p \sec a x - p \sec a x}\right) \ \mathrm d x} {q + p \sec a x}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 q \int \frac {\left({q + p \sec a x}\right) \ \mathrm d x} {q + p \sec a x} - \frac p q \int \frac {\sec a x \ \mathrm d x} {q + p \sec a x}\) Linear Combination of Integrals
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 q \int \mathrm d x - \frac p q \int \frac {\sec a x \ \mathrm d x} {q + p \sec a x}\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \frac x q - \frac p q \int \frac {\sec a x \ \mathrm d x} {q + p \sec a x} + C\) Primitive of Constant
\(\displaystyle \) \(=\) \(\displaystyle \frac x q - \frac p q \int \frac {\cos a x \, \sec a x \ \mathrm d x} {q \cos a x + p \cos a x \, \sec a x} + C\) multiplying top and bottom by $\cos a x$
\(\displaystyle \) \(=\) \(\displaystyle \frac x q - \frac p q \int \frac {\mathrm d x} {p + q \cos a x} + C\) Secant is Reciprocal of Cosine

$\blacksquare$


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