## Calculus (3rd Edition)

Published by W. H. Freeman

# Chapter 8 - Techniques of Integration - 8.7 Improper Integrals - Exercises - Page 441: 45

#### Answer

The integral diverges.

#### Work Step by Step

We have $\int_{-\infty}^{\infty} \dfrac{ x dx}{1+x^2} \ dx=\lim\limits_{R \to \infty} \int_{-\infty}^0\dfrac{ x dx}{1+x^2} \ dx \\=\lim\limits_{R \to \infty} \dfrac{1}{2} \ln (R^2+1) \\=\infty$ Also, $\int_0^\infty \dfrac{x dx}{1+x^2}=\lim\limits_{R \to \infty} \int_0^{R} \dfrac{x dx}{1+x^2}\\=\lim\limits_{R \to \infty} \dfrac{1}{2} \ln (R^2+1) \\=\infty$ Hence, the integral diverges.

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