Evaluate the integrals in Exercises 31–56. Some integrals do n...

Question

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.

∫ x² tan⁻¹(x / 2) dx

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 integral, the integral of x to the power of 2 multiplied by inverse tangent of x divided by 3 d x. OK, so it appears for this particular problem we're asked to evaluate the integral that is provided to us. So now that we know that we're ultimately trying to evaluate this integral. We need to note that in order to solve for the integral of x to the power of 2 multiplied by inverse tangent of x divided by 3 d x. Looking at this integral itself, we should be able to recall and recognize that we need to use integration by parts formula, which is, as we should recall, the integration by parts formula states the integral of UDV is equal to u multiplied by V minus the integral of V multiplied by DU. So with that in mind, our first step is we need to choose our values for U and DV. So let us say that for this problem, u will be equal to inverse tangent of x divided by 3. And we will now need to find the derivative with respect to x, so we need to find duu by dx, which is equal to 1 divided by 1 plus parentheses of x divided by 3. 32 multiplied by 1/3, which will be equal to 1 divided by 1 + x 2 divided by 9 multiplied by 1/3. Which will be, when we simplify down to its most simple form, will be equal to 3 divided by x to the power of 2 plus 9. I skipped a few steps, but essentially we're trying to write our final answers in its most simplified form. So rule of thumb, any anything you're solving for, make sure you can't simplify it anymore, and that means you'll have the correct answers when you can't simplify it anymore. So, with that said, we will note that DU will be equal to 3 divided by X2 + 9 DX. And let us state that DV will be equal to x 2DX. And we need to find the integral of both sides, so we need to take the integral of DV is equal to the integral of x 2. D X, which will mean that we will find that V will be equal to X to the power of 3 divided by 3. Fantastic, and that's when we perform the integration. So, moving right along here, our second step, so this is step 2, and our second step is we need to recall and apply the integration by parts formula. So we take the integral of x to the 2 multiplied by inverse tangent of X divided by 3. The x which will be equal to parentheses of x to the power of 3 divided by 3 multiplied by inverse tangent of x divided by 3 minus the integral of parentheses of x 3 divided by 3 multiplied by 3 divided by x 2 plus 9 d x. Fantastic. So with that said, what we need to do is we need to start our simplification. Which we'll find that this will be equal to x to the power of 3 divided by 3 multiplied by inverse tangent of x divided by 3 minus the integral of x 3 divided by x 2 + 9 dx. So now, What we need to do is we need to, for our third step, is we need to solve for the remaining integral, and in order to solve for the integral of x 3 divided by x2 + 9 dx, we will need to recall how to use simple algebraic manipulation. So what does that mean? So we need to take X 3 divided by X2 + 9. Which will be equal to x to the power of 3. Plus 9 x minus 9 X. Fantastic, divided by. X to the power of 2 + 9. Which will be equal to when we factor out our x x multiplied by parentheses of x 2 + 9 minus 9 x divided by x 2 + 9, fantastic, which will be all equal to x minus 9 x divided by x 2 + 9. So now, we need to integrate term by term, so we need to take the integral of parentheses of X minus 9 X divided by X2 + 9. DX will be equal to the integral of xDX minus 9 multiplied by the integral of x divided by x to the 2 + 9 DX. Fantastic. Moving right along here. So the first part is we need to note that the integral of XDX will be equal to x 2 divided by 2. And for our second integral, we will need to recall and use the substitution method, so we need to note thats will be equal to x 2 + 9, which will mean that ds by dx will be equal to 2x, which will mean that 1/2 multiplied by ds will be equal to 2 x dx. So with that in mind, we can state that 9 multiplied by the integral of x divided by x to the power of 2 plus 9 dx will be equal to 9 divided by 2 multiplied by the natural log of the absolute value of x 2 plus 9. And let's take a moment here to pause and note that x 2 is greater than or equal to 0, which will mean that x 2 + 9 is greater than or equal to 9, and because of this reason, we can remove the absolute values sign from our expression. So you can go ahead and write that 9 multiplied by the integral of x divided by x 2 + 9 DX will be equal to 9 divided by 2 multiplied by the natural log of parentheses of x 2 + 9. So, that will mean that the integral of x 3 divided by x 2 + 9 dx will be equal to x 2 divided by 2 minus 9 divided by 2 multiplied by the natural log of parentheses x 2 + 9. Fantastic. So our fourth and final step is we need to combine everything, so we need to substitute all of our results from step 3 back into our main equation. So you go ahead and write that x 3 divided by 3 multiplied by inverse tangent of x divided by 3 minus parentheses of x to the 2 divided by 2 minus 9 divided by 2. Multiplied by the natural log of parentheses of x 2 + 9 + c. Will be all equal to x 3 divided by 3 multiplied by inverse tangent of x divided by 3 minus x 2 divided by 2. Plus 9 divided by 2 multiplied by the natural log of parentheses of x to the 2 + 9 plus c where the c variable represents some constant variable and that's it. That is our final answer hooray, we did it. So going back to look at the problem itself, we could safely say that our final answer for our evaluated integral once again is without a doubt. X to the power of 3 divided by 3 multiplied by inverse tangent of x divided by 3 minus x to the power of 2 divided by 2 plus 9 divided by 2 multiplied by the natural log of x to the power of 2 plus 9 plus c and that's it. That is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.∫ x² tan⁻¹(x / 2) dx

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Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ x² tan⁻¹(x / 2) dx