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limit integral

Source: SEEMOUS 2014 P4

June 4, 2021
integrationlimitscalculus

Problem Statement

a) Prove that lim⁡n→∞n∫0narctan⁡xnx(x2+1)dx=π2\lim_{n\to\infty}n\int^n_0\frac{\operatorname{arctan}\frac xn}{x(x^2+1)}dx=\frac\pi2. b) Find the limit lim⁡n→∞n(m∫0narctan⁡xnx(x2+1)dx−π2)\lim_{n\to\infty}n\left(m\int^n_0\frac{\operatorname{arctan}\frac xn}{x(x^2+1)}dx-\frac\pi2\right).
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