Use integration by parts to obtain the formula ∫ √(1 - x²) dx ...

Question

Use integration by parts to obtain the formula ∫ √(1 - x²) dx = (1/2) x √(1 - x²) + (1/2) ∫ 1 / √(1 - x²) 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. Let i be equal to the integral of the square root of 4 minus x 2 dx. Using integration by parts, the integral can be written as the integral of the square root of 4 minus x to the 2 dx is equal to ax. Multiplied by the square root of 4 minus x 2 plus b multiplied by the integral of 1 divided by the square root of 4 minus x to the power of 2 d x. Find the values of the constants a and b. OK, so it appears for this particular prompt that we're asked to take all the information that is provided to us, and we're asked to ultimately determine what the values of the constants A and B are. So we're ultimately solving for two separate answers. What is A equal to and what is B equal to? So now that we know what we're solving for, we need to recall a note based on our prior knowledge. In order to solve this particular problem, we will need to recall and apply integration by parts by setting u be equal to the square root of 4 minus x 2. And dV be equal to dx, and then we will need to manipulate the resulting expression to reintroduce the original integral, and we will do this by algebraically rearranging the terms, and we will be utilizing the identity that states that x 2 is equal to 4 minus parentheses of 4 minus x 2. And we will therefore be able to isolate the integral and identify the constants A and B by comparing our result to the given form. So keeping our methodology in mind for how we're going to solve this problem, our first step that we will take is we will apply the integration by parts method. So let the integral be i is equal to the integral of the square root of 4 minus x to the power of 2 dx, and we will now need to recall and use the integration by parts formula, which states that the integral of UDV is equal to u multiplied by V minus the integral of VDU. And let us state that you will be equal to the square root of 4 minus x 2, which will mean that DU by DX will be equal to 1 divided by the square root. Or rather 1 divided by 2 multiplied by the square root of 4 minus x 2 multiplied by parentheses of 2x, which will mean that duu will be equal to x divided by the square root of 4 minus x to the 2 DX. And we need to let DV be equal to DX, which will mean that the integral of DV will be equal to the integral of DX, which will be meaning that V will be equal to X. So that's what the integral of DV is equal to the integral of DX means. It means that V is equal to X. So now we need to plug our values back into our formula, so we could write that i is equal to x multiplied by the square root of 4 minus x 2 minus the integral of x multiplied by g x divided by the square root of 4 minus x 2 dx. Which will mean that i will be equal to x multiplied by the square root of 4 minus x 2 plus the integral of x 2 divided by the square root of 4 minus x 2 dx. OK, our second step now is we need to manipulate the numerator. So in order to relate the integral back into its original form, as we should recall and recognize, we can rewrite. Once again, we're going to rewrite X 2 as parentheses of 4 minus X2 + 4. So when we apply this identity, we can go ahead and write that i is equal to x multiplied by the square root of 4 minus x 2 plus the integral of 4 minus parentheses of 4 minus x 2 divided by the square root of 4 minus x 2. The x, which will mean we can go ahead and write that i is equal to x multiplied by the square root of 4 minus x 2 plus the integral of negative parentheses of 4 minus x 2. Plus 4 divided by the square root of 4 minus x 2 dx. Our third step now is we need to split our integral, so we need to write i is equal to x multiplied by the square root of 4 minus x 2, plus the integral. Of, so it's gonna, it's gonna be minus actually. So, I is equal to x multiplied by the square root of 4 minus x 2 minus the integral of 4 minus x 2 divided by the square root of 4 minus x 2. DX plus the integral of 4 divided by the square root of 4 minus x to the power of 2 dx, and we need to note that since 4 minus x 2 divided by the square root of 4 minus x 2 is equal to the square root of 4 minus x 2. That will mean that our equation will become, as a result. I is equal to x multiplied by the square root of 4 minus x 2 minus the integral of the square root of 4 minus x 2 dx plus 4 multiplied by the integral of 1 divided by the square root of 4 minus x to the 2 dx. So for our fourth and final step, we now need to solve for i, and we need to notice that the integral of the square root of 4 minus x to the power of 2, d x is our original integral i. So we can go ahead and write that i is equal to x multiplied by the square root of 4 minus x 2. Minus 1 + 4 multiplied by the integral of 1 divided by the square root of 4 minus x to the power of 2 dx. We now need to move our I to the left side, so you will get 2i is equal to x multiplied by the square root of 4 minus x 2, plus 4 multiplied by the integral of 1 divided by the square root of 4 minus x to the power of 2. X. Fantastic. So what we need to do now is we need to isolate and solve peri. So we need to divide by 2 to get i by itself. So we will find that I is equal to 1 divided by 2 multiplied by X, multiplied by the square root of 4 minus. X to the power of 2 + 2 multiplied by the interval of 1 divided by the square root of 4 minus x to the power of 2 d x and now we need to compare this with i is equal to a multiplied by x multiplied by the square root of 4 minus x to the 2 plus b multiplied by the integral of 1 divided by the square root of. Or minus x to the power of 2 d x and that will mean as a result we will find that a will be simply equal to 1/2 and b will be simply equal to 2 and those are our final two answers hooray, we did it. So now that we solve for our final two answers, we now need to go back up to the top to recap what we've learned. So based on our prior knowledge of what we've learned by solving this problem, we could say that our final answers once again for the constants. A and B will be that once again, A is equal to 1/2, and B is equal to 2. And that's it. Those are our final two answers. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Use integration by parts to obtain the formula ∫ √(1 - x²) dx = (1/2) x √(1 - x²) + (1/2) ∫ 1 / √(1 - x²) dx.

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