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

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