Definite Integral to Infinity of Exponential of -a x minus Exponential of -b x over x by Secant of p x
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Theorem
- $\ds \int_0^\infty \frac {e^{-a x} - e^{-b x} } {x \sec p x} \rd x = \frac 1 2 \map \ln {\frac {b^2 + p^2} {a^2 + p^2} }$
where:
- $a$ and $b$ are non-negative real numbers
- $p$ is a real number.
Proof
Fix $p$ and set:
- $\ds \map I \alpha = \int_0^\infty \frac {e^{-\alpha x} } {x \sec p x} \rd x$
for all $\alpha \ge 0$.
Then:
- $\ds \int_0^\infty \frac {e^{-a x} - e^{-b x} } {x \sec p x} \rd x = \map I a - \map I b$
We have:
\(\ds \map {I'} \alpha\) | \(=\) | \(\ds \frac \d {\d \alpha} \int_0^\infty \frac {e^{-\alpha x} } {x \sec p x} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^\infty \frac \partial {\partial \alpha} \paren {\frac {e^{-\alpha x} } {x \sec p x} } \rd x\) | Definite Integral of Partial Derivative | |||||||||||
\(\ds \) | \(=\) | \(\ds -\int_0^\infty e^{-\alpha x} \cos p x \rd x\) | Derivative of $e^{a x}$, Definition of Secant Function | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac \alpha {\alpha^2 + p^2}\) | Definite Integral to Infinity of $e^{-a x} \cos b x$ |
so:
\(\ds \map I \alpha\) | \(=\) | \(\ds -\int \frac \alpha {\alpha^2 + p^2} \rd \alpha\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\frac 1 2 \map \ln {\alpha^2 + p^2} + C\) | Primitive of $\dfrac x {x^2 + a^2}$ |
for all $\alpha \ge 0$, for some constant $C \in \R$.
We then have:
\(\ds \int_0^\infty \frac {e^{-a x} - e^{-b x} } {x \sec p x} \rd x\) | \(=\) | \(\ds \map I a - \map I b\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-\frac 1 2 \map \ln {a^2 + p^2} + C} - \paren {-\frac 1 2 \map \ln {b^2 + p^2} + C}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \paren {\map \ln {b^2 + p^2} - \map \ln {a^2 + p^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \map \ln {\frac {b^2 + p^2} {a^2 + p^2} }\) | Difference of Logarithms |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Definite Integrals involving Exponential Functions: $15.87$