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Calculus IIChapter 1: Integration TechniquesSection 1-8

Improper Integrals — Question 10

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Question 10

Determine whether the integral converges. If it does, evaluate it. ∫0∞dx(1+x)2\int_0^\infty\frac{dx}{(1+x)^2}

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Question 10 – Solution

Step 1: Replace infinity with a limit. I=limb→∞∫0b(1+x)−2dx.I=\lim_{b\to\infty}\int_0^b(1+x)^{-2}\,dx. Step 2: Integrate. ∫(1+x)−2dx=−(1+x)−1=−11+x.\int(1+x)^{-2}\,dx=-(1+x)^{-1}=-\frac1{1+x}. Step 3: Evaluate the bounds and limit. I=limb→∞[−11+x]0b=limb→∞(−11+b+1)=0+1=1.\begin{align*} I&=\lim_{b\to\infty}\left[-\frac1{1+x}\right]_0^b\\ &=\lim_{b\to\infty}\left(-\frac1{1+b}+1\right)\\ &=0+1=1. \end{align*} 1 (convergent)\boxed{1\text{ (convergent)}}

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