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.

∫₀^π/2 x³ cos 2x 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. Compute the integral evaluated from 0 to pi divided by 2 of x to the power of 2 multiplied by sin of 2 x dx. OK, so it appears for this particular problem we're simply asked to compute or to solve for this integral that is provided to us by the prom itself. So now that we know what we're ultimately trying to solve for. We need to know based on our prior knowledge, in order to evaluate this definite integral, the integral evaluate from 0 toi divided by 2 of x to the 2 multiplied by sin of 2 x. DX. So in order to evaluate this definite integral, we will need to recall and use the method of integration by parts, and we will need to repeat this integration by parts method a couple times. We will need to repeat it until we're able to simplify our final answer for the integral. So we're gonna be using it in order to help us solve. For our integral. So with that said. Our first step that we need to take is we need to begin our integration by parts method. So let us say that you will be equal to x to the power of 2, which will mean that DU by Dx will be equal to 2x, which will mean that DU will be equal to 2 XDX. Which will mean that DV will be equal to sin of 2 xDX, which will be the interval of dV is equal to the interval of sin of 2 XDX, so 2 XDX, which will mean that V, so we'll sign that V will be equal to negative. 1/2 multiplied by cosine of 2 x. So now At this stage, we need to recall and apply the integration by Parks formula which states that the interval of UDV is equal to u multiplied by V minus the integral of V. DU 2 hour interval. So when we apply this integration by Parks formula, we will be able to write the integral evaluated from 0 to high divided by 2 of x to the power of 2. Multiplied by sin of 2 x. DX will be equal to square brackets of 1/2 multiplied by x2 multiplied by cosine of 2x. Evaluate from 0 to pi divided by 2. Minus So this will be minus. The integral evaluated from 0 to pi divided by 2 1/2 multiplied by cosine of 2 x multiplied by 2XDX. OK, we are on a roll here. So, moving right along here, we can go ahead and write that the integral evaluated from 0 to pi divided by 2 of x to the power of 2 multiplied by sin of 2 x. Dx will be equal to square brackets of 1/2 multiplied by x 2 multiplied by cosine of 2x. Evaluated from 0 to pi divided by 2, plus the integral evaluated from 0 to pi divided by 2 of x multiplied by cosine of 2 x dx. OK. Our second step now is we need to evaluate our boundary terms, so we need to take square brackets of 1/2 multiplied by x 2 multiplied by a cosine of 2 x. And we need to note that this will be evaluated from 02 I divided by 2. Which will be equal to 1/2 multiplied by pi divided by 2 to the power 2 multiplied by cosine of 2 multiplied by pi divided by. 2 minus 0 to 2 multiplied by cosine of 0. Which will be All equal to when we simplify our answer down to its most simple form, will be equal to pi to the power of 2 divided by 8. Fantastic. So, our third step, so this is step number 3, is we need to apply integration by parts method again. So with that in mind, in order to evaluate the integral evaluated from 0 to 1 divided by 2 of x multiplied by cosine of 2 x dx, we will need to apply our integration by Park's method again. So for this particular interval, we will state that U is equal to x, which will mean that DU by DX will be equal to 1, which will mean that DU will be simply equal to DX. And that will mean that DV is equal to cosine of 2 XDX will be, will give us that the integral of DV will be equal to the integral of cosine of 2 x. D X will mean that V. Will be simply equal to 1/2 multiplied by sin of 2 x. OK. We are on a roll. Thus, so when we apply the integration by parts formula, we can go ahead and write that the interval evaluated from 0 to pi divided by 2 of x multiplied by cosine of 2 x dx will be equal to square brackets of x multiplied by 1/2 multiplied by sin of 2 x. Evaluated from 0 to pi divided by 2 minus the integral evaluated from 0 to pi divided by 2 of 1/2 multiplied by sin of 2 x dx. Fantastic. So, we need to note that our expression. Now will become equal to square brackets of Uh, so it's gonna be square brackets of X divided by 2 multiplied by sin of 2 x. And then we need to note that this will be evaluated from 0 to pi divided by 2. -1/2 multiplied by square brackets of 1/2 multiplied by cosine of 2 x evaluated from 0 to pi divided by 2, OK. Which will give us that it's equal to 1/2 multiplied by square brackets of x multiplied by sin of 2 x evaluated from 0 to pi divided by 2. Plus 1 divided by 4 multiplied by square brackets of cosine of 2 x evaluate from 0 to pi divided by 2. Which will be equal to 1/2 multiplied by square brackets of i divided by 2 multiplied by sin of pi. Minus. 1/2 multiplied by square brackets of 0 multiplied by sin of 0. Plus 1/4 multiplied by square brackets of cosine of pi minus 1 divided by 4 multiplied by cosine of 0. So when we evaluate this expression, we will find that it's simply equal to -152. So for our fourth step, we now need to combine all of our results, and when we do, we will find that the integral evaluated from 0 to pi divided by 2 of x to the 2 multiplied by sin of 2 x dx will be simply equal to i divided, so it's gonna be pi to the power of 2 divided by 8. Minus 1/2, which we can also write this expression as i to the power of 2 minus 4 all divided by 8, and that's it. That is our final answer hooray, we did it. So looking at the prom itself, we can safely say once again without a doubt that our evaluated computed integral is once again our final answer is pi to the power of 2 minus 4 divided by 8, and that's it. That's 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.∫₀^π/2 x³ cos 2x dx

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Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫₀^π/2 x³ cos 2x dx