Primitive of Reciprocal of p squared plus square of q by Cosine of a x
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Theorem
- $\ds \int \frac {\d x} {p^2 + q^2 \cos^2 a x} = \frac 1 {a p \sqrt{p^2 + q^2} } \arctan \frac {p \tan a x} {\sqrt {p^2 + q^2} } + C$
where $C$ is an arbitrary constant.
Proof
\(\ds \int \frac {\d x} {p^2 + q^2 \cos^2 a x}\) | \(=\) | \(\ds \int \frac {\csc^2 a x \rd x} {p^2 \csc^2 a x + q^2 \cot^2 a x}\) | multiplying the numerator and the denominator by $\csc^2 a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\csc^2 a x \rd x} {p^2 + \paren {p^2 + q^2} \cot^2 a x}\) | Difference of Squares of Cosecant and Cotangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {-\paren {\cot a x}' \rd x} {a p^2 + a \paren {p^2 + q^2} \cot^2 a x}\) | Derivative of Cotangent Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {-\d t} {a p^2 + a \paren {p^2 + q^2} t^2}\) | putting $t = \cot a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-1} {a \paren {p^2 + q^2} } \int \frac {\d t} {\paren {\tfrac p {\sqrt {p^2 + q^2} } }^2 + t^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-1} {a \paren {p^2 + q^2} } \frac {\sqrt {p^2 + q^2} } p \, \map \arctan {\frac {\sqrt {p^2 + q^2} } p t } + C\) | Primitive of $\dfrac 1 {x^2 + a^2}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-1} {a p \sqrt {p^2 + q^2} } \map \arctan {\frac {\sqrt {p^2 + q^2} } p t} + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-1} {a p \sqrt {p^2 + q^2} } \map \arctan {\frac {\sqrt {p^2 + q^2} } p \cot a x} + C\) | substituting $t = \cot a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-1} {a p \sqrt {p^2 + q^2} } \map \arctan {\frac {\sqrt {p^2 + q^2} } {p \tan a x} } + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-1} {a p \sqrt {p^2 + q^2} } \map \arccot {\frac {p \tan a x} {\sqrt {p^2 + q^2} } } + C\) | Arctangent of Reciprocal equals Arccotangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a p \sqrt {p^2 + q^2} } \paren {\map \arctan {\frac {p \tan a x} {\sqrt {p^2 + q^2} } } - \frac \pi 2} + C\) | Sum of Arctangent and Arccotangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a p \sqrt {p^2 + q^2} } \map \arctan {\frac {p \tan a x} {\sqrt {p^2 + q^2} } } + C\) | absorbing into the arbitrary constant |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\cos a x$: $14.392$