Primitive of Reciprocal of Cosine of a x by 1 plus Sine of a x/Weierstrass Substitution
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Lemma for Primitive of Reciprocal of $\cos a x \paren {1 + \sin a x}$
The Weierstrass Substitution of $\ds \int \frac {\d x} {\cos a x \paren {1 + \sin a x} }$ is:
- $\ds \frac 2 a \int \frac {\d u} {\paren {1 - u} \paren {1 + u}^3}$
where $u = \tan \dfrac {a x} 2$.
Proof
\(\ds \int \frac {\d x} {\cos a x \paren {1 + \sin a x} }\) | \(=\) | \(\ds \frac 1 a \int \frac {\d z} {\cos z \paren {1 + \sin z} }\) | Primitive of Function of Constant Multiple: $z = a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \int \frac {\dfrac {2 \rd u} {1 + u^2} } {\dfrac {1 - u^2} {1 + u^2} \paren {1 + \dfrac {2 u} {1 + u^2} } }\) | Weierstrass Substitution: $u = \tan \dfrac z 2 = \tan \dfrac {a x} 2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 a \int \frac {\d u} {\paren {1 - u^2} \paren {1 + u^2 + 2 u} }\) | multiplying top and bottom by $1 + u^2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 a \int \frac {\d u} {\paren {1 - u^2} \paren {1 + u}^2}\) | Square of Sum | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 a \int \frac {\d u} {\paren {1 - u} \paren {1 + u} \paren {1 + u}^2}\) | Difference of Two Squares | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 a \int \frac {\d u} {\paren {1 - u} \paren {1 + u}^3}\) | simplifying |
$\blacksquare$