Definite Integral from 0 to 2 Pi of Reciprocal of Square of a plus b Cosine x
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Theorem
- $\ds \int_0^{2 \pi} \frac {\d x} {\paren {a + b \cos x}^2} = \frac {2 \pi a} {\paren {a^2 - b^2}^{3/2} }$
where $a$ and $b$ are real numbers with $a > b > 0$.
Proof 1
From Definite Integral from $0$ to $2 \pi$ of $\dfrac 1 {a + b \cos x}$, we have:
- $\ds \int_0^{2 \pi} \frac {\d x} {a + b \cos x} = \frac {2 \pi} {\sqrt {a^2 - b^2} }$
We have:
\(\ds \frac \partial {\partial a} \int_0^{2 \pi} \frac {\d x} {a + b \cos x}\) | \(=\) | \(\ds \int_0^{2 \pi} \frac \partial {\partial a} \paren {\frac 1 {a + b \cos x} } \rd x\) | Definite Integral of Partial Derivative | |||||||||||
\(\ds \) | \(=\) | \(\ds -\int_0^{2 \pi} \frac {\d x} {\paren {a + b \cos x}^2}\) | Quotient Rule for Derivatives |
and:
\(\ds \frac \partial {\partial a} \paren {\frac {2 \pi} {\sqrt {a^2 - b^2} } }\) | \(=\) | \(\ds 2 \pi \paren {\frac {-2 a} {2 \sqrt {a^2 - b^2} } } \paren {\frac 1 {a^2 - b^2} }\) | Quotient Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac {2 \pi a} {\paren {a^2 - b^2}^{3/2} }\) |
giving:
- $\ds \int_0^{2 \pi} \frac {\d x} {\paren {a + b \cos x}^2} = \frac {2 \pi a} {\paren {a^2 - b^2}^{3/2} }$
$\blacksquare$
Proof 2
\(\ds \int_0^{2 \pi} \frac {\d x} {\paren {a + b \cos x}^2}\) | \(=\) | \(\ds \intlimits {\frac {b \sin x} {\paren {b^2 - a^2} \paren {a + b \cos x} } } 0 {2 \pi} - \frac a {b^2 - a^2} \int_0^{2 \pi} \frac {\d x} {a + b \cos x}\) | Primitive of Reciprocal of square of p plus q by Cosine of a x | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {b \sin 2 \pi} {\paren {b^2 - a^2} \paren {a + b \cos 2 \pi} } - \frac {b \sin 0} {\paren {b^2 - a^2} \paren {a + b \cos 0} } + \frac a {a^2 - b^2} \int_0^{2 \pi} \frac {\d x} {a + b \cos x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac a {a^2 - b^2} \int_0^{2 \pi} \frac {\d x} {a + b \cos x}\) | Sine of Integer Multiple of Pi | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac a {a^2 - b^2} \times \frac {2 \pi} {\sqrt {a^2 - b^2} }\) | Definite Integral from 0 to 2 Pi of Reciprocal of a plus b Cosine x | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 \pi a} {\paren {a^2 - b^2}^{3/2} }\) |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Definite Integrals involving Trigonometric Functions: $15.46$