Primitive of Reciprocal of 1 minus Sine of a x/Corollary 2

Theorem

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

Proof

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

\(=\)

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

Primitive of $\dfrac 1 {1 - \sin x}$: Tangent form

\(\ds \)

\(=\)

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

setting $a \gets 1$

\(\ds \)

\(=\)

\(\ds \)

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Sources

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