Evaluate the integrals in Exercises 1–22.
∫₀^π 8cos⁴(2πx...

Question

Evaluate the integrals in Exercises 1–22.

∫₀^π 8cos⁴(2πx) 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 1 of 12 multiplied by cosine to the power of 4 of pi multiplied by xdx. OK, so it appears for this particular problem, we're asked to compute or to solve for the integral that is provided to us. So now that we know that we're ultimately trying to compute this integral. Our first step that we need to take based on our prior knowledge, we need to be able to recognize that in order to evaluate the definite integral, the integral evaluated from 0 to 1 of 12 multiplied by cosine to the power of 4 of. Pi multiplied by XDX, we will need to expand the integrand and apply the substitution method. So by looking at our integral, we need to be able to recognize that we will need to expand the integrand and we will need to recall and apply the substitution method. So with that said, our first step is, is we need to expand the integrand by recalling and using the following identities. So firstly, we need to recall and use the power reduction identity that states that cosine squared of theta is equal to 1 plus cosine of 2 theta divided by 2. Thus we can go ahead and write that cosine to the power of 4 of pi multiplied by x. Will be equal to drum roll parentheses of cosine squared of pi multiplied by x. To the power 2, which will be equal to parentheses of 1 plus cosine of 2 multiplied by pi multiplied by x, all divided by 2, all to the power of 2. Fantastic. So what we need to do now is we need to expand the square. So in order to do that, we need to take cosine to the power of 4 of pi multiplied by x is going to be equal to 1 divided by 4 multiplied by parentheses of 1 + 2 multiplied by cosine of 2 multiplied by pi multiplied by x plus. Cosine to the power of 2 of 2 multiplied by pi multiplied by x. OK. So now, at this stage, we need to, once again, we need to recall and apply the identity again to, we're applying the identity to cosine squared of 2 multiplied by pi multiplied by X is equal to 1 plus cosine of, so we're going to be applying it to 1 plus cosine of. Or multiplied by pi multiplied by x, all divided by 2. So with that in mind, we can go ahead and write that cosine to the power of 4 of pi multiplied by x is equal to 1 divided by 4 multiplied by parentheses of 1 plus 2 multiplied by cosine of. 2 multiplied by pi multiplied by x. Plus, and then we're gonna add 1 plus cosine of 4 multiplied by pi multiplied by X, all divided by drum roll 2. So it's gonna be all divided by 2. Fantastic. And this is going to be. All equal to 1 divided by 4 multiplied by parentheses of 1 + 2 multiplied by cosine of 2 multiplied by pi multiplied by x plus 1 divided by 2. Plus cosine of 4 multiplied by pi multiplied by x divided by 2. And this will be equal to 1/4 or 1 divided by 4 multiplied by 3 divided by 2, plus 2 multiplied by cosine of 2 multiplied by pi multiplied by x plus cosine of or multiplied by pi multiplied by x, all divided by 2. And finally, this will be equal to. 3 divided by 8 multiply or it's gonna be 3 divided by 8 plus 1 divided by 2 multiplied by cosine of 2 multiplied by pi multiplied by X plus. 1/8 or 1 divided by 8 multiplied by cosine of 4 multiplied by pi multiplied by x. Fantastic. So now we need to multiply by the constant 12. So we need to take 12 multiplied by cosine to the power of 4 of pi multiplied by x, which will be equal to 9 divided by 2, plus 6 multiplied by cosine of 2 multiplied by pi multiplied by x plus. 3 divided by 2 multiplied by cosine of 4 multiplied by pi multiplied by x. Fantastic. For our second step, we need to set up the definite integral, and in order to do that based on our prior knowledge, we will need to split the integral into 3 separate parts. So we need to write the integral evaluated from 0 to 1 of 9 divided by 2, dx plus the integral evaluated from 0 to 1 of 6 multiplied by cosine. Of 2 multiplied by pi multiplied by X D X. And then we need to add. The integral, so we're gonna be adding the integral evaluated from 0 to 1 of 3 divided by 2. Multiplied by cosine of 4 multiplied by pi multiplied by x dx. Fantastic. So our third step now is we're going to be evaluating our first term. So we're going to evaluate the integral evaluated from 0 to 1 of 9 divided by 2, DX, which will be equal to square brackets of 9 divided by 2, multiplied by x, evaluated from 0 to 1, which will be equal to 9 divided by 2. Multiplied by 1 minus 9 divided by 2 multiplied by 0 and this will give us an answer that is equal to 9 divided by 2. Fantastic. So moving right along here. Our fourth step is we need to evaluate our second term, so we need to take the integral evaluated from 0 to 1 of 6 multiplied by cosine of 2 multiplied by pi multiplied by XDX, and in order to solve for this term or this integral. Expressively, we need to use the substitution method. So we're calling and using the substitute substitution method. Let us state that you will be equal to 2 multiplied by pi multiplied by x. That will thenfore mean that DU, so we can say that DU by DX, so DU by DX will be equal to 2 pi, which will mean that DX will be equal to DU divided by 2 multiplied by pi. Fantastic. So now, at this point, we need to change the limits. So if we change the limits, if X is equal to 0, that will mean that you multiply. So once again, that will mean that you will be equal to 2 pi, so you will be equal to. 2 pi multiplied by 0, which will be equal to 0, and if x is equal to 1. That will mean that you will be equal to 2 pi multiplied by 1, which will be equal to 2 pi. Fantastic. So that means we can go ahead and write that the integral, so once again we're writing that if the integral evaluated using our change limits, evaluated from 0 to 2 pi. Of 6 multiplied by cosine of u multiplied by duu by 2 pi or DU divided by 2 pi will be equal to 3 divided by pi multiplied by the integral evaluated from 0 to 2 pi of cosine of u. DU which will be equal to 3 divided by pi. Multiplied by the square brackets of sin of u. Evaluated from 0 to 2 pi. And this will be equal to 3 divided by pi multiplied by parentheses of sin of 2 pi minus sin of 0, which will be equal to 3 divided by pi multiplied by parentheses of 0 minus 0. Which will be equal to 0. Fantastic. And finally, for our step number 5, we need to evaluate our third term, our third integral. So for the integral, evaluate from 0 to 1 of 3 divided by 2 multiplied by cosine of. 4 pi or 4 multiplied by pi multiplied by x dx, we need to recall and use the substitution method again. So let us state that V will equal to V will be equal to 4 multiplied by pi multiplied by X. Then that will mean that DV by DX will be equal to 4 pi, and that will mean that DX will be equal to DV divided by 4 multiplied by pi. So, now we need to change the limits. So if X is equal to 0, that will mean that V is equal to 4 multiplied by 0, which will be equal to 0. And if x is equal to 1, that will mean that V is equal to 4 pi multiplied by 1, which will be equal to 4 pi. So using our changed limits, we can write that the integral evaluated from 0 to 4 pi of 3 divided by 2 multiplied by cosine of v multiplied by dv divided by 4 pi. Will be equal to 3 divided by 8 pi multiplied by the integral evaluated from 0 to 4 pi of cosine of VDV. And this will be equal to 3 divided by 8 pi multiplied by square brackets of sin of, so sin of. V, so this will be the sign of V. Evaluated from 0 to 4 high. And this will be equal to 3 divided by 8 multiplied by pi, so 3 divided by 8 pi multiplied by parentheses of. Sign of or pi minus sin of 0. Which will be equal to 3 divided by 8 pi multiplied by parentheses of 0 minus 0, which will be equal to 0. So our sixth and final step is we need to combine our results. So summing the values of the three parts, we will have 9 divided by 2 plus 0 + 0, which will be equal to 9 divided by 2. So that will mean that our final answer without a doubt, the integral evaluated from 0 to 1 of 12 multiplied by cosine to the power of 4 of pi multiplied by x, dx will be equal to 9 divided by 2, and that's it. That is our final answer. Hooray, we did it. So going back to review the problem itself, we can safely say that our final answer, once again for this, or that when we compute the integral, will be, once again, 9 divided by 2. 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 1–22.∫₀^π 8cos⁴(2πx) dx

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Evaluate the integrals in Exercises 1–22.
∫₀^π 8cos⁴(2πx) dx