Question

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

Accepted Answer

Recognize that the integral is an improper integral because the lower limit is negative infinity. This means we need to express the integral as a limit: $$\displaystyle \int_{-\infty}^2 \frac{2}{x^2$ + 4} \, dx = \lim_{a 	o -\infty} \int_a^2$ \frac{2}{x^2$ + 4} \, dx. $$Recall the standard integral formula for $$\int \frac{dx}{x^2$ + a^2$} = \frac{1}{a} \arctan\left(\frac{x}{a}ight) + C. In $$this problem, $$a^2 = 4$, so a = 2$. Rewrite the integrand to match the formula: $$\frac{2}{$$x^2 + 4$$} = 2 \cdot \frac{1}{$$x^2$$ + $$2^2$$}$$. This allows us to factor out the constant 2 and apply the formula. Evaluate the definite integral from a$ to 2 using the antiderivative: 2$$ \cdot \left[ \frac{1}{2} \arctan\left( \frac{x}{2} ight) ight]_$$a^2$$ = \left[ \arctan\left( \frac{x}{2} ight) ight]_$$a^2. $$Take the limit as $$a$$ approaches negative infinity: $$\lim_{a 	o -\infty} \left( \arctan\left( \frac{2}{2} ight) - \arctan\left( \frac{a}{2} ight) ight). $$Use the fact that $$\arctan(x)$$ approaches $$-\frac{\pi}{2} as$$ $$x 	o -\infty to $$complete the evaluation.

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

Verified video answer for a similar problem:

Key Concepts

Improper Integrals

Integration of Rational Functions

Limits and Convergence of Integrals