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Evaluate the integrals$$\int 8 \cos ^{4} 2 \pi x d x$$

Calculus 1 / AB

Calculus 2 / BC

Chapter 8

Techniques of Integration

Section 2

Trigonometric Integrals

Integrals

Integration

Integration Techniques

Harvey Mudd College

Baylor University

Idaho State University

Lectures

01:11

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Section eight that to problem number 18. So I'm looking from the anti derivative Eighth Coast onto the 4th 2 Pi X. So, first of all, the key to most of your signing cousin is together in terms of Cho Sun's graders sign square because then I can use my identity's the Pythagorean dignity or the coastline double angle so I can write. This is eight. Co sign squared to Pi X squared. I know that the co sign squared of X is equal to 1/2 one plus the co signer to X. That tells me that the co sign squared of two pi acts is equal to 1/2 one plus the co sign of four pi X so I can rewrite this integral as eight and this is going to be 1/2 one plus co sign of four pi X squared DX and sorry about this. Make sure there's a common mistake that I almost made here. What is being squared is all of this. Okay, so this was substituting for that particular terms. All of that is being squared. So this becomes two ah times the integral of one plus co sign of four pi X squared DX. Okay, so now let's just rewrite what that is. So two times Integral one plus co sign of four pi X squared the X that is going to be equal to two and then I'm gonna have one plus to co sign for pi X plus co sign squared four pi axe DX And now I can use that same trick Identity Here, this is going to be 1/2 one plus co sign of eight pi x So I end up with two times the interval one plus to co sign for pi X plus when half one plus co sign eight pi x the ex Okay, And so this becomes s O you add the one and the 1/2 and that's three have. So this is too. And then you have Ah huh s So you've got sorry. Three plus to co sign four pi X. So I added them with one in the 1/2 there. So that's become so. That's three halves and sorry. So that is three halves to cosign on four pi x plus and you're gonna have 1/2 co sign a pi x dx. Okay, and now all of this would need to be integrated. So let's just figure out what that is. Let's go and distribute the two across everything in this interval. So the first time you're gonna have a threes, you're gonna have interval of three. And then the two times the two co signed four Pi x is four co sign for pie axe, and then you're going to have the two times of 1/2. So you're gonna have plus co sign of a pi x plus co sign eight pi x all of this DX. And so when I integrate this, I'm gonna end up with three ex and then the integral of the coastline is gonna be the sine function. You're gonna have four sign four pi X divided by for pie plus And then you have the sign of eight pi X divided by a pie plus a constant of integration. So your answer is three x plus one over pi signed four pi eggs plus 1/8 High sign of eight pi x plus a constant of integration. So again, what do we do? This one? The key here is when you see coast onto the fourth. I'm gonna need to organize that in terms of co sign squared. Then I had to use my double angle formula for co sign. Not only that, I have to use it once, but I had to use that twice every time I saw co sign squared to convert it to a co sign of a double angle. Then it was a matter of me not making arithmetic mistakes. There was a couple of pauses there when I made sure of that and then powering through.

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