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

Question

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

∫₀^∞ (16 tan⁻¹x dx) / (1 + x²)

Accepted Answer

Hello there. Today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Evaluate the improper integral. The integral evaluated from 1 to positive infinity of. 8 multiplied by inverse tangent of x all divided by 1 plus x to the power of 2 dx. OK, so it appears for this particular prompt we are given an improper integral and were asked to evaluate this improper integral. So now that we know that we're ultimately asked to evaluate this improper integral that is given to us. Our first step that we need to take, so step number 1, based on our prior knowledge, is we need to rewrite this integral with a limit. So what does that mean? So that means we need to take the integral evaluated from 1 to positive infinity of 8 multiplied by inverse tangent of x, all divided by 1 plus x to the power of 2. The x which will be equal to the limit as b approaches positive infinity of the integral of 1 or rather the integral evaluated from 1 to b. Of 8 multiplied by inverse tangent of x divided by 1 plus x to the power of 2 dx. OK. Our second step now is we need to determine or find the antiderivative of the integral of 8 multiplied by inverse tangent of x divided by 1 + x to the power of 2 dx. So we are ultimately asked to solve for the antiderivative of this integral. So in order to do that, let us use, or rather, let us recall and use the substitution method. So let us state that you will be equal to inverse tangent of X. So that will mean that DU by DX will be equal to 1 divided by 1 plus x to the power of 2. Which will mean that DU will be equal to 1 divided by 1 + X to the power of 2 DX. Fantastic. So with that in mind, we can go ahead and write that the integral of 8 multiplied by inverse tangent of x divided by 1 + x to the power of 2 dx will be equal to 8 multiplied by the integral of u duu, which will be equal to 8 multiplied by parentheses of u to the 2 divided by 2. Plusc where c denotes some constant variable which will be equal to 4 multiplied by u to the power 2 plus c which will be equal to 4 multiplied by parentheses of inverse tangent of x to the power of 2. Plus, so it's gonna be once again, inverse tangent of X to the power of 2. Plus C, fantastic. It means we can't forget our constant variable. So, moving right along here, our third step, so this is step 3, and we need to, at this point, evaluate the definite integral. So, once it, with that in mind, we're evaluating our definite integral now, so we need to take the integral evaluated from 1 to B of. 8 multiplied by inverse tangent of x divided by 1 plus x to the power of 2 d x, which will be equal to 4 multiplied by parentheses of inverse tangent of x to the power of 2 evaluated from 1 to b. And this will be equal to or multiplied by square brackets of parentheses of inverse tangent of b to the power of 2 minus parentheses inverse tangent of 1 to the power of 2. Which will be equal to 4 multiplied by square brackets of parentheses of inverse tangent of b to the 2 minus parentheses pi divided by 4 to the power of 2. Which will mean that it will be equal to 4 multiplied by parentheses of inverse tangent of b to the power of 2 minus 4 multiplied by pi to the power of 2 divided by 16. Which will be equal to 4 multiplied by parentheses of inverse tangent of b to the 2 minus pi to 2 divided by 4. Fantastic. We are on a roll here. So our fourth and final step, this is step 4, is we need to take the limit as b approaches positive infinity. And we need to recall that as B approaches positive infinity, that will mean that inverse tangent of B will approach pi divided by 2. So, with that said, we can go ahead and write that the limit as B approaches positive infinity of. Parentheses of 4 multiplied by parentheses of inverse tangent of B, so inverse tangent of B to the power of 2. Minus. I to the power of 2 divided by 4 will be equal to 4 multiplied by parentheses of pi divided by 2 to 2 minus pi 2 divided by 4, which will be equal to pi to 2 minus pi to the power of 2 divided by 4. Which shall be equal to when we write our final answer in its most simplified form 3 multiplied by pi to the power of 2 divided by 4, and this is our correct and final answer. Hooray, we did it. So going back to the top to look at the problem itself and to recap what we've learned, we could safely say that our evaluated improper integral will be. 3 multiplied by pi to the power 2 divided by 4, and this is our correct and final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.∫₀^∞ (16 tan⁻¹x dx) / (1 + x²)

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The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀^∞ (16 tan⁻¹x dx) / (1 + x²)