### 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.