# Primitive of Reciprocal of Sine of a x by Square of Cosine of a x

## Theorem

$\displaystyle \int \frac {\mathrm d x} {\sin a x \cos^2 a x} = \frac 1 a \ln \left\vert{\tan \frac {a x} 2}\right\vert + \frac 1 {a \cos a x} + C$

## Proof

 $\displaystyle \int \frac {\mathrm d x} {\sin a x \cos^2 a x}$ $=$ $\displaystyle \int \frac {\left({\sin^2 a x + \cos^2 a x}\right) \ \mathrm d x} {\sin a x \cos^2 a x}$ Sum of Squares of Sine and Cosine $\displaystyle$ $=$ $\displaystyle \int \frac {\sin^2 a x \ \mathrm d x} {\sin a x \cos^2 a x} + \int \frac {\cos^2 a x \ \mathrm d x} {\sin a x \cos^2 a x}$ Linear Combination of Integrals $\displaystyle$ $=$ $\displaystyle \int \frac {\mathrm d x} {\sin a x} + \int \frac {\sin a x \ \mathrm d x} {\cos^2 a x}$ simplifying $\displaystyle$ $=$ $\displaystyle \int \csc a x \ \mathrm d x + \int \frac {\sin a x \ \mathrm d x} {\cos^2 a x}$ Cosecant is Reciprocal of Sine $\displaystyle$ $=$ $\displaystyle \int \csc a x \ \mathrm d x + \int \frac {\tan a x \ \mathrm d x} {\cos a x}$ Tangent is Sine divided by Cosine $\displaystyle$ $=$ $\displaystyle \int \csc a x \ \mathrm d x + \int \sec a x \tan a x \ \mathrm d x$ Secant is Reciprocal of Cosine $\displaystyle$ $=$ $\displaystyle \frac 1 a \ln \left\vert{\tan \frac {a x} 2}\right\vert + \int \sec a x \tan a x \ \mathrm d x + C$ Primitive of $\csc a x$ $\displaystyle$ $=$ $\displaystyle \frac 1 a \ln \left\vert{\tan \frac {a x} 2}\right\vert + \frac 1 {a \cos a x} + C$ Primitive of $\sec^n a x \tan a x$

$\blacksquare$