Primitive of Reciprocal of Square of Sine of a x

Theorem

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

Corollary

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

Proof

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

\(=\)

\(\ds \int \csc^2 a x \rd x\)

Definition of Cosecant

\(\ds \)

\(=\)

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

Primitive of $\csc^2 a x$

$\blacksquare$

Also see

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

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$: $14.351$