Primitive of Reciprocal of Sine of x by Cosine of x

Theorem

$\ds \int \frac {\d x} {\sin x \cos x} = \ln \size {\tan x} + C$

Proof

From Primitive of $\dfrac 1 {\sin a x \cos a x}$:

$\ds \int \frac {\d x} {\sin a x \cos a x} = \frac 1 a \ln \size {\tan a x} + C$

The result follows by setting $a = 1$.

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals