Definite Integral to Infinity of Power of x by Logarithm of x over One plus x

Theorem

$\ds \int_0^\infty \frac {x^{p - 1} \ln x} {1 + x} \rd x = -\pi^2 \csc p \pi \cot p \pi$

where:

$p$ is a real number with $0 < p < 1$

$\csc$ denotes the cosecant function

$\cot$ denotes the cotangent function.

$\ds \int_0^\infty \frac {x^{p - 1} \ln x} {1 + x} \rd x$

$=$

$\ds \int_0^\infty \frac 1 {1 + x} \map {\frac \partial {\partial p} } {x^{p - 1} } \rd x$

$\ds $

$=$

$\ds \frac \d {\d p} \int_0^\infty \frac {x^{p - 1} } {1 + x} \rd x$

$\ds $

$=$

$\ds \map {\frac \d {\d p} } {\pi \csc p \pi}$

$\ds $

$=$

$\ds -\pi^2 \csc p \pi \cot p \pi$

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Definite Integrals involving Logarithmic Functions: $15.97$