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Chapter 12: Problem 20

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{-\infty}^{0} \frac{x}{x^{2}+1} d x $$

Short Answer

Expert verified
The integral converges and equals to \(蟺/2\).

Step by step solution

01

Substitution

First, instead of directly dealing with -鈭�, let鈥檚 put a variable b in place of -鈭�. Now, the integral becomes \(\int_{b}^{0} \frac{x}{x^{2}+1} dx\). At the end, we鈥檒l take the limit of b approaching -鈭�.
02

Evaluate the Integral

To get the integral, one could recognize that the integrand is the derivative of \(arctan(x)\), hence \(\int_{b}^{0} \frac{x}{x^{2}+1} dx = [arctan(x)]_{b}^0\).
03

Find the limit as b approaches -鈭�

After finding the antiderivative of the integrand, substitute the bounds of the integral back in: \(= arctan(0)-arctan(b)\). Now, take the limit of this as \(b\) approaches \(-鈭瀄). This becomes: \(\lim_{b \to -\infty} [arctan(0) - arctan(b)] = 0 - -蟺/2 = 蟺/2\).

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Most popular questions from this chapter

Profit The net profits \(P\) (in billions of dollars per year) for The Hershey Company from 2002 through 2005 can be modeled by \(P=\sqrt{0.00645 t^{2}+0.1673}, \quad 2 \leq t \leq 5\) where \(t\) is time in years, with \(t=2\) corresponding to 2002 . Find the average net profit over that time period. (Source: The Hershey Co.)

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