Exercise 30, § 1 Functions of a Real Variable, III, 123 a) If mmm and nnn are two integers such that 0<m<n0 < m < n0<m<n show that ∫0+∞xm−11+xndx=πnsinmπn. \int_0^{+\infty} \frac{x^{m-1}}{1 + x^n} dx = \frac{\pi}{n \sin \frac{m \pi}{n}}. ∫0+∞1+xnxm−1dx=nsinnmππ. b) Show that for 0<a<10 < a < 10<a<1 the integral ∫0+∞xa−11+xdx\int_0^{+\infty} \frac{x^{a-1}}{1 + x} dx∫0+∞1+xxa−1dx is uniformly convergent for aaa varying in a compact interval, and deduce from a) that ∫0+∞xa−11+xdx=πsinaπ. \int_0^{+\infty} \frac{x^{a-1}}{1 + x} dx = \frac{\pi}{\sin a \pi}. ∫0+∞1+xxa−1dx=sinaππ. Solution No solution has been written yet.