Definite Integral to Infinity of Arctangent of p x minus Arctangent of q x over x
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Theorem
- $\ds \int_0^\infty \frac {\arctan p x - \arctan q x} x \rd x = \frac \pi 2 \ln \frac p q$
where $p$ and $q$ are strictly positive real numbers.
Proof
Note that the integrand is of the form:
- $\ds \int_0^\infty \frac {\map f {p x} - \map f {q x} } x \rd x$
where:
- $\map f x = \arctan x$
We have, by Derivative of Arctangent Function:
- $\map {f'} x = \dfrac 1 {1 + x^2}$
which is continuous on $\R$.
By Limit to Infinity of Arctangent Function:
- $\ds \lim_{x \mathop \to \infty} \map f x = \lim_{x \mathop \to \infty} \arctan x = \frac \pi 2$
As $f$ is continuously differentiable and $\ds \lim_{x \mathop \to \infty} \map f x$ exists and is finite, we may apply Frullani's Integral, giving:
\(\ds \int_0^\infty \frac {\arctan p x - \arctan q x} x \rd x\) | \(=\) | \(\ds \paren {\lim_{x \mathop \to \infty} \arctan x - \arctan 0} \ln \frac p q\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac \pi 2 \ln \frac p q\) | Arctangent of Zero is Zero |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Definite Integrals involving Trigonometric Functions: $15.67$