Primitive of Reciprocal of p plus q by Sine of a x/Weierstrass Substitution

Lemma for Primitive of Reciprocal of $p + q \sin a x$

The Weierstrass Substitution of $\ds \int \frac {\d x} {p + q \sin a x}$ is:

$\ds \frac 2 a \int \frac {\d u} {p u^2 + 2 q u + p}$

where $u = \tan \dfrac {a x} 2$.

Proof

\(\ds \int \frac {\d x} {p + q \sin a x}\)

\(=\)

\(\ds \frac 1 a \int \frac {\d z} {p + q \sin z}\)

Primitive of Function of Constant Multiple: $z = a x$

\(\ds \)

\(=\)

\(\ds \frac 1 a \int \frac 1 {p + q \frac {2 u} {u^2 + 1} } \frac {2 \rd u} {u^2 + 1}\)

Weierstrass Substitution: $u = \tan \dfrac z 2 = \tan \dfrac {a x} 2$

\(\ds \)

\(=\)

\(\ds \frac 1 a \int \frac {2 \rd u} {\paren {u^2 + 1} \frac {p \paren {u^2 + 1} + 2 q u} {u^2 + 1} }\)

common denominator

\(\ds \)

\(=\)

\(\ds \frac 2 a \int \frac {\d u} {p u^2 + 2 q u + p}\)

simplifying

$\blacksquare$