Primitive of Reciprocal of p plus q by Tangent of a x
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Theorem
- $\ds \int \frac {\d x} {p + q \tan a x} = \frac {p x} {p^2 + q^2} + \frac q {a \paren {p^2 + q^2} } \ln \size {q \sin a x + p \cos a x} + C$
Proof 1
First, let $\arctan \dfrac p q = \phi$.
Let $z = a x + \phi$.
\(\ds z\) | \(=\) | \(\ds \map \sin {a x + \phi}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d z} {\d x}\) | \(=\) | \(\ds a \map \cos {a x + \phi}\) | Derivative of $\sin a x$ etc. | ||||||||||
\(\ds \) | \(=\) | \(\ds a \cos z\) |
Then:
\(\ds \int \frac {\d x} {p + q \tan a x}\) | \(=\) | \(\ds \int \frac {\d x} {p + q \dfrac {\sin a x} {\cos a x} }\) | Tangent is Sine divided by Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\cos a x \rd x} {p \cos a x + q \sin a x}\) | multiplying top and bottom by $\cos a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\cos a x \rd x} {\sqrt {p^2 + q^2} \map \sin {a x + \phi} }\) | Multiple of Sine plus Multiple of Cosine: Sine Form | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {\sqrt {p^2 + q^2} } \int \frac {\cos a x \rd x} {\map \sin {a x + \phi} }\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {\sqrt {p^2 + q^2} } \paren {\frac {\ln \size {\map \sin {a x + \phi} } } {a \cos \phi} + \tan \phi \int \frac {\sin a x \rd x} {\map \sin {a x + \phi} } + C}\) | Primitive of $\dfrac {\cos a x} {\map \sin {a x + \phi} }$ |
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Proof 2
We have:
- $\dfrac \d {\d x} \paren {q \sin a x + p \cos a x} = a q \cos a x - a p \sin a x$
Thus:
\(\ds \int \frac {\d x} {p + q \tan a x}\) | \(=\) | \(\ds \int \frac {\d x} {p + q \dfrac {\sin a x} {\cos a x} }\) | Tangent is Sine divided by Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\cos a x \rd x} {p \cos a x + q \sin a x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {p^2 + q^2} \int \frac {\paren {p^2 + q^2} \cos a x \rd x} {p \cos a x + q \sin a x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {p^2 + q^2} \int \frac {p^2 \cos a x + p q \sin a x - p q \sin a x + q^2 \cos a x} {p \cos a x + q \sin a x} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {p^2 + q^2} \paren {\int \frac {p^2 \cos a x + p q \sin a x} {p \cos a x + q \sin a x} \rd x + \int \frac {-p q \sin a x + q^2 \cos a x} {p \cos a x + q \sin a x} \rd x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {p^2 + q^2} \paren {\int p \rd x + \frac q a \int \frac {\map \d {p \cos a x + q \sin a x} } {p \cos a x + q \sin a x} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {p x} {p^2 + q^2} + \frac q {a \paren {p^2 + q^2} } \ln \size {p \cos a x + q \sin a x} + C\) | Primitive of Constant and Primitive of Reciprocal |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\tan a x$: $14.438$