Video: Integrating Cosine Squared

Determine ∫ 8 cos² 𝑥 d𝑥.

02:24

Video Transcript

Determine the integral of eight cos squared 𝑥.

So, the first thing I’ve done is I’ve taken outside the eight, which is just a constant. So, now what we’ve got is eight multiplied by the integral of cos squared 𝑥. So, now the next step is to apply a product-to-sum formula. And the one that’s gonna be useful to us is that cos squared 𝑥 equals a half cos two 𝑥 plus one.

So then, we have eight multiplied by the integral of a half cos two 𝑥 plus one. So, now again what I’m gonna do is take out the constant. So, we’re gonna take the half out. So, we’re gonna have eight multiplied by a half, which is just four. So, now we have four multiplied by the integral of cos two 𝑥 plus one.

And when we’re integrating, we can integrate each part separately. So, if we look at integrating cos two 𝑥, well, we should know what the result of this is. But I was just gonna show you how it is if we wanted to use substitution to do it. Well, first of all, if we say that 𝑢 is equal two 𝑥, then d𝑢 d𝑥 will be equal to two. So, therefore, d𝑥 will be equal to a half d𝑢.

So, now we’d have a half multiplied by the integral of cos 𝑢. So, this would be equal to a half sin 𝑢. And that’s because it’s just one of our standard integrals. Cos 𝑢, if we integrate, is gonna be sin 𝑢. And then, all we do is substitute in 𝑢 is equal to two 𝑥. And we can see that the result is a half sin two 𝑥. Okay, so, that’s that part integrated.

And now the second part, which should be nice and straightforward. So, we want to integrate one. And when we integrate one, we just get 𝑥. It’s worth noting that I haven’t put the constant of integration on here because I’m gonna do that at the end when we put it all together. So, let’s put everything back in.

So, what we get is four multiplied by a half sin two 𝑥 plus 𝑥 and then plus 𝑐, which takes us to our final answer. So, we can say that the integral of eight cos squared 𝑥 is equal to four 𝑥 plus two sin two 𝑥 plus 𝑐.

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