Primitive of Reciprocal of 1 plus Sine of a x/Corollary

Theorem

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

Proof

\(\ds \int \frac {\d x} {1 + \sin a x}\)

\(=\)

\(\ds -\frac 1 a \map \tan {\frac \pi 4 - \frac {a x} 2} + C\)

Primitive of $\dfrac 1 {1 + \sin a x}$

\(\ds \leadsto \ \ \)

\(\ds \int \frac {\d x} {1 + \sin x}\)

\(=\)

\(\ds -\map \tan {\frac \pi 4 - \frac x 2} + C\)

setting $a \gets 1$

\(\ds \)

\(=\)

\(\ds \map \tan {\frac x 2 - \frac \pi 4} + C\)

Tangent Function is Odd

$\blacksquare$

Sources

• 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals