Primitive of Reciprocal of Square of Sine of a x/Corollary

Corollary to Primitive of $\dfrac 1 {\sin^2 a x}$

$\ds \int \frac {\d x} {\sin^2 x} = -\cot x + C$

Proof

\(\ds \int \frac {\d x} {\sin^2 a x}\)

\(=\)

\(\ds \frac {-\cot a x} a + C\)

Primitive of $\dfrac 1 {\sin^2 a x}$

\(\ds \leadsto \ \ \)

\(\ds \int \frac {\d x} {\sin^2 x}\)

\(=\)

\(\ds \frac {-\map \cot {1 \times x} } 1 + C\)

setting $a \gets 1$

\(\ds \)

\(=\)

\(\ds -\cot x + C\)

simplifying

$\blacksquare$

Sources

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