Evaluate ∫ x³ √(1 - x²) dx using:
a. Integration by part...

Question

Evaluate ∫ x³ √(1 - x²) dx using:

a. Integration by parts.

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 3 multiplied by the square root of 9 + x 2 dx. OK, so it appears for this particular prompt, we are asked to evaluate the integral that is provided to us. So with that in mind, in order to evaluate this integral, which once again is the integral of x 3 multiplied by the square root of 9 + x 2 dx, by looking at the integral itself, we should be able to recognize based on our prior knowledge that we can use integration by parts by strategically splitting the x to the power of 3 term. So with that in mind, since we are going to use the integration by parts method, our first step that we will need to take in order to implement the integration by parts method is we need to identify our values or variables for u and dV. So we need to assign u and dV. So with that said, we can split, so let's make a note here, we can split x to the power of 3. Into x to the power of 2 multiplied by x so that the dv part is easily integrable, meaning we can integrate it using a simple substitution. So with that in mind, let us state that u will be equal to x to the power of 2. And that DV will be equal to x multiplied by the square root of x plus. X to the power of 2 DX. Fantastic. So, moving right along here. Our 2nd step now is we need to calculate DU and V. So we need to find our variables for D U and V. So starting with DU, we can say that DU will be equal to, so once again, DU will be equal to 2 xDX, and to solve for V, V is a little bit more complicated. So in order to find V, we need to note that V will be equal to the integral of x multiplied by the square root of. 9 plus x to the power of 2 dx and to solve for v, we will need to use the substitution method. So we're calling and using the substitution method. We will state that w will be equal to 9 plus x to the power of 2. Which will give us that DW will be equal to 2 XDX. Fantastic. So that will mean that we can go ahead and write that V will be equal to 1/2 multiplied by the integral of w to the power of 1/2 DW, which will be equal to 1/2 multiplied by 2 divided by 3. Multiplied by w to the power of 3 divided by 2, which will be equal to 1/3 multiplied by parentheses of 9 plus x to the 2 in parentheses to the power of 3 divided by 2. Fantastic. So now our third step is we need to recall and apply our integration by parts formula, which states that the integral of UDV is equal to u multiplied by V, so you. Multiplied by V is going to be subtracted from the integral of V multiplied by DU. So, once again, our integration by parts formula is the integral of UDV is equal to u multiplied by V minus the integral of VDU. So, applying this formula, we can go ahead and write that the integral of x 3 multiplied by the square root of 9 + x 2 DX will be equal to, so drum roll, so this will be. Equal to x to the 2 multiplied by square brackets of 1 divided by 3 multiplied by parentheses of 9 plus x to the power of 2, the power of 3 divided by 2. Minus the integral of 1/3 multiplied by 9 plus x 2 to the power of 3 divided by 2 fantastic, and this will be multiplied by parentheses 2 x dx. Switch will mean that it will be when we simplify a little bit, there will be equal to x 2 divided by 3 multiplied by parentheses of 9 plus x to the power of 2 in parentheses to the power of 3 divided by 2. -2 divided by 3 multiplied by the integral of x multiplied by parentheses of 9 + x to the power of 2. And this will be to the power of 3 divided by 2 DX. Fantastic. So now, moving right along here. Our fourth step now is we need to evaluate the remaining integral. So in order to solve for the integral of x multiplied by parentheses of 9 + x 2 to the power of 3 divided by 2, the x, we will need to use the substitution method again. So we need to state that w will be equal to 9 plus x 2. And that DW will be equal to 2 XDX, which is the same substitution as we used earlier, if you should recognize. OK. So with that said, we can go ahead and apply our substitution method so we can write 2 divided by 3 multiplied by the integral of x multiplied by parentheses of 9 plus x to the 2 to the power of 3 divided by 2 dx. Will be equal to 2 divided by 3 multiplied by 1/2 multiplied by the integral of w to the power of 3 divided by 2 dw. Which will be equal to 1/3 multiplied by 2/5 multiplied by w to the power of 5 divided by 2, which will be, when we simplify, be equal to 2 divided by 15 multiplied by parentheses of 9 plus x 2 to the power of 5 divided by 2. So now for our 5th and final step, so this will be step number 5, our last step, is we need to combine our terms and we need to add the constant value or the constant variable c. So we can go ahead and write that the integral of x to the power of 3 multiplied by the square root of 9 plus x to the power of 2. The x will be equal to x 2 divided by 2 multiplied by parentheses of 9 + x 2 entheses the power of 3 divided by 2 minus 2 divided by 15 multiplied by parentheses of 9 + x to the power of 2. To the power of 5 divided by 2 plus c. And to simplify our factored form, in order, once again, we need to make sure we simplify your answer down to its most simple form, so we need to factor out 1 divided by 15 multiplied by parentheses of 9 plus x to the power of 2. To the power of 3 divided by 2. So with that said, we can go ahead and write parentheses 9. Plus X to the power of 2. To the power of 3 divided by 2 divided by 15 multiplied by square brackets of 5 x 2 minus 2 multiplied by parentheses of 9 plus x to the power of 2 plus c. And when we manipulate this expression down to its most simple form, which I'll skip a few steps, we just need to use some algebra, you will find that our final answer written in its most simplified form is 1/5 multiplied by parentheses of 9 + x 2 to the power of 3 divided by 2. Multiplied by x 2 minus 6 + c. And that's it. That is our final answer. Hooray, we did it. So going back to the top to recap what we've learned, we can safely say based on what we've learned together that our final answer once again for our evaluated integral is as follows. 1/5 multiplied by parentheses of 9 + x 2 to the power of 3 divided by 2, multiplied by parentheses of x 2 minus 6 + C, where once again C represents some constant variable. And that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Evaluate ∫ x³ √(1 - x²) dx using:a. Integration by parts.

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Evaluate ∫ x³ √(1 - x²) dx using:
a. Integration by parts.