Question

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.∫₋∞^∞ (x dx) / (x² + 4)^(3/2)

Accepted Answer

Identify the integral to be evaluated: $$\displaystyle \int_{-\infty}^{\infty} \frac{x}{(x$$^{2}$$ + 4$$)^{3/2}$$} \, dx. $$Observe the integrand function $$f(x) = \frac{x}{(x$$^{2}$$ + 4$$)^{3/2}$$}. $$Notice that the numerator is an odd function in $$x ($$since $$x is $$odd) and the denominator is an even function (depends on $$x^2). $$Therefore, the entire integrand is an odd function because the quotient of an odd function by an even function is odd. Recall that the integral of an odd function over symmetric limits $$[-a, a] is $$zero, provided the integral converges. Since the limits here are $$-\infty to$$ $$\infty$$, which are symmetric about zero, and the integral converges, the integral evaluates to zero. Thus, without performing any complicated integration, conclude that the value of the integral is zero due to the odd symmetry of the integrand over symmetric limits. If you want to verify convergence, consider the behavior of the integrand as $$x 	o \infty$$: the denominator grows like $$x^{3}$$, and the numerator grows like $$x$$, so the integrand behaves like $$\frac{x}{x$$^{3}$$} = \frac{1}{x$$^{2}$$}$$, which is integrable over $$[1, \infty)$$, confirming convergence.

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋∞^∞ (x dx) / (x² + 4)^(3/2)

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Key Concepts

Improper Integrals

Symmetry of Functions

Substitution Method