The integrals in Exercises 1–34 converge. Evaluate the integra...

Question

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

∫₋∞² (2 dx) / (x² + 4)

Accepted Answer

Welcome back everyone. Evaluate the improper integral from 0 to infinity of 4 dx divided by x2 + 16. Let's go ahead and rewrite the integral. We have integral from 0 to infinity of 4 D X. Divided by x2 + 16. Now since this is an improper integral, we're going to rewrite it in terms of a limit. Limit of the integral from 0 to b, we are replacing infinity with b as b approaches infinity of 4 d x divided by x2 + 16. Followed by the X. And now let's focus on the integral. We're going to call our limit L, and considering the limit. And in particular the integral that we got within the limit, so that's integral from 0 to b of we can factor out 4, right, because it's a constant, and we are integrating d x divided by x 2 plus 16. Now let's analyze this integral. Specifically, we can rewrite 16s 4 squared, and we can remember that the integral of d x divided by x2 plus a 2 is equal to 1 divided by a. Inverse of tangent of x divided by a plus a constant of integration c. So in this case, we're going to get 4 multiplied by 1 divided by 4, because our A is 4. Inverse of tangent of x divided by 4, no constant of integration is required because we're evaluating from zero to b, right? So we're gonna cancel out 4 1/4. And notice that our limit l becomes. Limit as b approaches infinity. Of Inverse of tangent. Applying the fundamental theorem of calculus part 2, we first of all get b divided by 4 minus inverse of tangent of 0 divided by 4. Now inverse of tangent of 0 is 0 and now as b approaches infinity. We're going to get inverse of tangent of an infinitely large number, right? And in particular. Inverse of tangent of an infinitely large number is going to give us a limit of pi divided by 2. So we get a limit of 5 divided by 2, which is also the value of the integral. So that's our final answer, and thank you for watching.

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

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