Primitive of Reciprocal of 1 plus Sine of a x

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Theorem

$\ds \int \frac {\d x} {1 + \sin a x} = -\frac 1 a \map \tan {\frac \pi 4 - \frac {a x} 2} + C$


Corollary

$\ds \int \frac {\d x} {1 + \sin x} = \map \tan {\frac x 2 - \frac \pi 4} + C$


Proof

\(\ds \int \frac {\d x} {1 + \sin a x}\) \(=\) \(\ds \int \frac 1 2 \map {\sec^2} {\frac \pi 4 - \frac {a x} 2} \rd x\) Reciprocal of One Plus Sine
\(\text {(1)}: \quad\) \(\ds \) \(=\) \(\ds \frac 1 2 \int \map {\sec^2} {\frac \pi 4 - \frac {a x} 2} \rd x\) Primitive of Constant Multiple of Function


Then:

\(\ds z\) \(=\) \(\ds \frac \pi 4 - \frac {a x} 2\)
\(\ds \frac {\d z} {\d x}\) \(=\) \(\ds -\frac a 2\) Derivative of Power
\(\ds \leadsto \ \ \) \(\ds \int \frac {\d x} {1 + \sin a x}\) \(=\) \(\ds -\frac 1 2 \int \frac 2 a \sec^2 z \rd z\) Integration by Substitution from $(1)$
\(\ds \) \(=\) \(\ds -\frac 1 2 \cdot \frac 2 a \tan z + C\) Primitive of Square of Secant Function
\(\ds \) \(=\) \(\ds -\frac 1 a \map \tan {\frac \pi 4 - \frac {a x} 2} + C\) substituting for $z$

$\blacksquare$


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