Primitive of Reciprocal of p by Sine of a x plus q by Cosine of a x
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Theorem
- $\ds \int \frac {\d x} {p \sin a x + q \cos a x} = \frac 1 {a \sqrt {p^2 + q^2} } \ln \tan \size {\frac {a x + \arctan \dfrac q p} 2} + C$
Proof
Lemma
- $\ds \frac 1 2 \map \arctan {\dfrac {-p} q} + \frac \pi 4 = \frac {\arctan \dfrac q p} 2$
$\Box$
\(\ds \int \frac {\d x} {p \sin a x + q \cos a x}\) | \(=\) | \(\ds \int \frac {\d x} {\sqrt {p^2 + q^2} \map \cos {a x + \arctan \dfrac {-p} q} }\) | Multiple of Sine plus Multiple of Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {\sqrt {p^2 + q^2} } \int \map \sec {a x + \arctan \dfrac {-p} q} \rd x\) | Secant is Reciprocal of Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a \sqrt {p^2 + q^2} } \int \sec z \rd z\) | Primitive of Function of $a x + b$: $z = a x + \arctan \dfrac {-p} q$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a \sqrt {p^2 + q^2} } \ln \tan \size {\frac z 2 + \frac \pi 4} + C\) | Primitive of $\sec a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a \sqrt {p^2 + q^2} } \ln \tan \size {\frac 1 2 \paren {a x + \arctan \dfrac {-p} q + \frac \pi 2} } + C\) | substituting for $z$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a \sqrt {p^2 + q^2} } \ln \tan \size {\frac {a x + \arctan \dfrac q p} 2} + C\) | Lemma |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.419$