Definite Integral to Infinity of Exponential of -a x minus Exponential of -b x over x
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Theorem
- $\ds \int_0^\infty \frac {e^{-a x} - e^{-b x} } x \rd x = \ln \frac b a$
where $a$ and $b$ are strictly positive real numbers.
Proof
Note that the integrand is of the form:
- $\ds \int_0^\infty \frac {\map f {a x} - \map f {b x} } x \rd x$
where:
- $\map f x = e^{-x}$
We have, by Derivative of Exponential Function:
- $\map {f'} x = -e^{-x}$
which is continuous on $\R$.
We also have, by Exponential Tends to Zero and Infinity:
- $\ds \lim_{x \mathop \to \infty} \map f x = \lim_{x \mathop \to \infty} e^{-x} = 0$
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:
\(\ds \int_0^\infty \frac {e^{-a x} - e^{-b x} } x \rd x\) | \(=\) | \(\ds \paren {\lim_{x \mathop \to \infty} e^{-x} - e^0} \ln \frac a b\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\ln \frac a b\) | Exponential of Zero | |||||||||||
\(\ds \) | \(=\) | \(\ds \ln \frac b a\) | Logarithm of Reciprocal |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Definite Integrals involving Exponential Functions: $15.71$