Primitive of Reciprocal of p plus q by Sine of a x/Weierstrass Substitution
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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 |
$\blacksquare$