Many methods needed Show that in the following steps.a.
Chapter 7, Problem 85AE(choose chapter or problem)
Many methods needed Show that \(\int_{0}^{\infty} \frac{\sqrt{x} \ln x}{(1+x)^{2}} \ d x=\pi\) in the following steps.
a. Integrate by parts with \(u=\sqrt{x} \ln x\).
b. Change variables by letting y =1/x.
c. Show that \(\int_{0}^{1} \frac{\ln x}{\sqrt{x}(1+x)} d x=-\int_{1}^{\infty} \frac{\ln x}{\sqrt{x}(1+x)} \ d x\) and conclude that \(\int_{0}^{\infty} \frac{\ln x}{\sqrt{x}(1+x)} \ d x=0\).
d. Evaluate the remaining integral using the change of variables \(z=\sqrt{x}\).
(Source: Mathematics Magazine 59, no. 1, February 1986: 49)
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