For \( a > b > 0 \), find the value of
\[
\int_0^{\infty} \frac{e^{ax} – e^{bx}}{x(e^{ax}+1)(e^{bx}+1)} dx.
\]

Prove or disprove that for all positive integers \(m\) and \(n\), \[ f(m,n)=\frac{2^{3(m+n)-\frac12} }{{\pi}} \int_0^{\pi/2} \sin^{ 2n – \frac12 }\theta \cdot \cos^{2m+\frac12}\theta \, d\theta\]  is an integer.

(A typo is fixed on Saturday.)

Compute \(\int_0^1 \frac{x^k-1}{\log x}dx\).

Compute \[ f(x)= \int_{0}^1 \frac{\log (1- 2t\cos x + t^2) }{t} dt.\]