Evaluate the integrals in Exercises 53–58.
∫ from -π/2 t...

Question

Evaluate the integrals in Exercises 53–58.

∫ from -π/2 to π/2 of cos(x) cos(7x) 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 integral evaluated from negative pi divided by 2 to positive pi divided by 2. Of cosine of 2x multiplied by cosine of 6 XDX. OK. So it appears for this particular prompt, we're asked to evaluate the integral that is provided to us by the prompt itself. So now that we know that we're ultimately trying to solve for the integral, let us note that our first step that we will need to take is we need to let i be equal to the integral evaluated from pi divided by 2 to pi divided by 2. Of cosine of 2 x multiplied by cosine of 6 XDX. And now, at this point, we need to recall and use the product to sum identity, which states that, as we should recall, cosine of a multiplied by cosine of B is equal to 1 divided by 2 multiplied by parentheses of. Cosine of a minus B plus cosine of A + B. Fantastic. Thus, we can go ahead and write that cosine of 2 x multiplied by cosine of 6 x will be equal to 1 divided by 2 multiplied by parentheses of cosine of 2 x minus 6x plus cosine of 2x plus 6 x. Which will be equal to 1 divided by 2 multiplied by parentheses of cosine of -4 x plus cosine of 8X. So we need to recall and note based on our prior knowledge that the cosine equation, so this cosine function will be even. So for any real argument T, we will have, as we should note. Cosine of negative t is equal to cosine of t. And taking T is equal to 4 X will give us. Will give us the value of cosine of 4 x, which will be equal to cosine of or x. So once again, T equals 4X will give us cosine of -4 X is equal to cosine of 4 X. So therefore we can go ahead and write that 1 divided by 2 multiplied by parentheses of cosine of -4 x plus cosine of 8x will be equal to 1 divided by 2 multiplied by parentheses of. 4 x plus cosine of. 8 X. So now we need to substitute our values back into our integral, and then we will need to evaluate. And when we do that, we will find that i is equal to the integral evaluated from pi divided by 2 to pi divided by 2 of 1 divided by 2 multiplied by cosine of or x plus cosine of. 8 X D X. Fantastic. Which will mean we can go ahead and write that i is equal to 1 divided by 2 of parentheses of the integral evaluated from pi divided by 2 to pi divided by 2 of cosine of 4 x dx plus the integral. Evaluated from negative pi divided by 2 to i divided by 2 of cosine of 8 XDX. Fantastic. And we need to note that since cosine of 4 x and cosine of 8x are even, we can therefore recall and apply the even function property formula, which states that the integral evaluated from negative A to positive A of. F XDX is equal to 2 multiplied by the integral evaluated from 0 to A F XDX can be applied. So with that in mind. We can go ahead and write that i is equal to 1/2 or 1 divided by 2 multiplied by 2 multiplied by parentheses of the integral evaluated from 0 to pi divided by 2 of cosine of 4 x d x. Plus the integral evaluated from 0 to pi divided by 2 of cosine of 8 XDX. Which means that we go ahead and write that i is equal to 1 divided by 4 multiplied by the integral evaluated from 0 to pi divided by 2 of. Cosine of 4 x multiplied by d of 4 x plus 1 divided by 8 multiplied by the integral evaluated from 0 to pi divided by 2 of cosine of 8x multiplied by d of 8 x. Fantastic. So, we can therefore, based on that expression, we can go ahead and write that I is equal to 1 divided by 4 multiplied by sin of 4 x, which is evaluated from 0 to pi divided by 2 plus 1 divided by 8. Multiplied by sin of 8 x evaluated from 0 to pi divided by 2. Which will mean that i is equal to 1 divided by 4 multiplied by parentheses of sin of 4 multiplied by pi divided by 2 minus sin of 0. And we need to add. 1 divided by 8 multiplied by parentheses of. Sign of, so it's going to be parentheses of sin of 8 multiplied by pi divided by 2. Minus of 0. So, when we simplify a bit, we can go ahead and write that I is equal to 1 divided by 4 multiplied by sin of 2 pi. Plus 1 divided by 8 multiplied by sin of 4 pi. Which will mean we can go ahead and write that I is equal to 1 divided by 4 multiplied by 0 plus 1 divided by 8 multiplied by 0, which will mean that I, when we evaluate that expression, will be simply equal to 0 and 0 is our correct and final answer. Hooray, we did it. So going back to look at the prom itself, we can safely say without a doubt that our final answer for the evaluated integral once again is simply 0, 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 the integrals in Exercises 53–58.∫ from -π/2 to π/2 of cos(x) cos(7x) dx

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Evaluate the integrals in Exercises 53–58.
∫ from -π/2 to π/2 of cos(x) cos(7x) dx