Use the table of integrals at the back of the text to evaluate...

Question

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.

∫ x arctan(x) 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 interval of x multiplied by inverse cosine of XDX. OK. So it appears for this particular prompt, we're ultimately asked to evaluate or rather solve for the integral that is provided to us by the prompt itself. So now that we know that we're ultimately trying to solve for this integral. Our first step that we need to take is we need to recall that in order to evaluate the integral of x multiplied by inverse cosine of xDX, we need to note, based on our prior knowledge that we can recall and use a standard formula for the product of a power of x and an inverse trigonometric function. And we need to note that for any integer n cannot equal -1, that will mean that the general integral formula for xn multiplied by inverse cosine of x will be drum roll, the integral of x to the power of n, so, the integral of x to the power of n. Multiplied by inverse cosine of x dx, which will be equal to x to the power of n + 1 divided by n + 1 multiplied by inverse cosine of x plus 1 divided by n + 1. Multiplied by the interval of x to the power of n + 1. Divided by the square root of 1 minus x 2. DX Fantastic. So, moving right along here. So when we apply this formula, we need to note that for the case of our problem. N will be equal to 1, so let us plug in our values into our formula template. So when we substitute n is equal to 1, we will be able to write that the integral of x multiplied by inverse cosine of x dx will be equal to. X 2 divided by 2 multiplied by inverse cosine. Of x plus 1/2 multiplied by the interval of x 2 divided by the square root of 1 minus x 2 d x. And we now need to evaluate the remaining interval. And the integral of x 2 divided by the square root of 1 minus x 2. DX is a standard form and usually found in integer tables, so this, you could easily identify it and locate it in an integral table. And we will find that. The integral of x 2 divided by the square root of 1 minus x 2 dx will be equal to 1/2 multiplied by parentheses of inverse sin of x minus x so it's gonna be minus x multiplied by the square root of 1 minus x 2. Plus C1. And we now need to combine the terms, saying a moment that c represents some constant integer. So when we combine terms, we could go ahead and write that x 2 divided by 2 multiplied by inverse cosine of x plus 1/2 multiplied by the integral of x 2 divided by the square root of 1 minus x 2. DX will be equal to x 2 divided by 2 multiplied by inverse cosine of x, so inverse cosine of x. And then we will need to add plus 1/2 multiplied by square brackets of 1/22 multiplied by Parentheses of which I'm gonna move down here, so it's gonna be square brackets of 1/2 multiplied by parentheses of inverse sin of x minus x multiplied by the square root of 1 minus x 2. Plus C1 and don't forget your other square bracket. And this will be equal to x 2 divided by 2 multiplied by inverse cosine of x plus 1/4 multiplied by parentheses of inverse sin of x minus x multiplied by the square root of 1 minus x to the power of 2 plus 1/2 multiplied by c1. Fantastic. So moving right along here, that will mean that this is all equal to x 2 divided by 2 multiplied by inverse cosine of x plus 1/4 multiplied by parentheses of inverse sin of x minus x multiplied by the square root of 1 minus x to the power of 2. In parentheses, plus C. And we need to note. Where C is equal to 1/2 multiplied by C1. So that will mean that our final answer without a doubt will be. Once again, X 2 divided by 2 multiplied by inverse cosine of x plus 1/4 multiplied by parentheses of inverse sin of x minus x multiplied by the square root of 1 minus x 2 + c. Fantastic. So going back up to the top to recap what we've learned, that will mean that our final answer once again, without a doubt, will be X to the power of 2 divided by 2 multiplied by inverse cosine of x plus 1 divided by 4 multiplied by parentheses of inverse sin of x minus x multiplied by the square root of 1 minus x to the power of 2. 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.

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.∫ x arctan(x) dx

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Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ x arctan(x) dx