Limit of a Definite Integral #5

Calculus Level 4

$\large{L=\lim_{n \to \infty} \int_{-n}^{n} \dfrac{dx}{(x^2+1)(e^x+1)}}$

If $L$ can be expressed as ${\dfrac{\pi^A}{B}}$ where $A,B$ are positive integers, then find the value of $A+B$ .

1 solution

Vishwak Srinivasan
Jul 21, 2015

Using the property,

$\displaystyle \int_{-a}^{a} f(x) dx = \int_{0}^{a} f(x) + f(-x) dx$

Here $f(x) = \dfrac{1}{(x^2 +1)(e^x + 1)}$

$f(x) + f(-x) = \dfrac{1}{(x^2 + 1)(e^x + 1)} + \dfrac{e^x}{(x^2 + 1)(e^x + 1)} = \dfrac{1}{x^2 + 1}$

$\Rightarrow L = \displaystyle \lim_{n \to \infty} \int_{0}^{n} \dfrac{dx}{x^2 + 1}$

$\Rightarrow L = \displaystyle \lim_{n \to \infty} \tan^{-1}n = \dfrac{\pi}{2}$

$A = 1 , B = 2 \Rightarrow A + B = \boxed{3}$

Simple standard approach.