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The integral$$ \int_0^\infty \frac{1}{\sqrt{x} (1 + x)}\ dx $$is improper for two reasons: The interval $ [0, \infty) $ is infinite and the integrand has an infinite discontinuity at 0. Evaluate it by expressing it as a sum of improper integrals of Type 2 and Type 1 as follows:$$ \int_0^\infty \frac{1}{\sqrt{x} (1 + x)}\ dx = \int_0^1 \frac{1}{\sqrt{x} (1 + x)}\ dx + \int_1^\infty \frac{1}{\sqrt{x} (1 + x)}\ dx $$
$\pi$
Calculus 2 / BC
Chapter 7
Techniques of Integration
Section 8
Improper Integrals
Integration Techniques
Missouri State University
Idaho State University
Boston College
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