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[music] With all that we've learned so
far, it's super easy to differentiate sine

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squared x, so I'm going to differentiate
sine squared.

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Well, it's the chain rule, right?
It's the derivative of the outside

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function, which is the squaring function.
So, it's 2 times the inside function,

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which is sine x, times the derivative of
the inside, which is cosine x.

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So there, that's the derivative of sine
squared x.

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What if I wanted to go backwards?
What if I wanted to anti-differentiate

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sine squared x?
I want to anti-differentiate sine squared

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x.
I'm looking for a function whose

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derivative is sine squared x.
Well, me and my first guess for central

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function might be sine cubed over 3.
Because if I differentiate this while I'm

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differentiating the outside function,
which is cubing, so it's 3 times the

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inside function squared divided by 3, and
that's looking good.

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There, I've got a sine squared x.
The problem is that, now I've got to

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multiply this by the derivative of the
inside function, which is sine, and the

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derivative of sine is cosine.
So, the derivative of this is not sine

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squared x, it's sine squared x times this
extra cosine.

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Oh, it seems really hard.
Well, there's a so-called half angle

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identity.
This is the trick.

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That sine squared x is, according to half
angle identity, 1 minus cosine of 2x all

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over 2.
Well, that's true.

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But how does that identity help?
Well, since sine squared x is 1 minus

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cosine 2x all over 2, that means the
anti-derivative for sine squared x is

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equal to the anti-derivative of 1 minus
cosine 2x over 2.

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Now to do this, this is the
anti-derivative of 1 half times something.

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So, I can pull out that factor of 1 half
and just figure out how to

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anti-differentiate 1 minus cosine of 2x.
Now, this is an anti-derivative of a

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difference, which is the difference of
anti-derivatives.

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So, I want to anti-differentiate 1 and I
want to anti-differentiate cosine 2x.

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Now, what's an anti-derivative of 1?
Well, it's just x.

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And what's an anti-derivative of cosine
2x?

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Well, one such anti-derivative is sine of
2x.

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But then, I've got to divide by 2 to
compensate for the chain rule here.

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It'll add plus c.
So, there we go.

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I could simplify this a little bit.
I could write this as x over 2 minus sine

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2x over 4 plus c, and there is my general
anti-derivative for sine x.

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Maybe you really like this sort of thing.
Well, if you are really having fun with

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this kind of thing, here's a little
challenge that you can play with.

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You can try to find an anti-derivative
say, for sine to the 4th, right?

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And if you don't like the number 4, you
can make the 4 some other number, right?

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But it's pretty fun.
Then, you can try to apply some of the

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trig identities to replace sine to some
power with things that you can currently

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anti-differentiate.
It's honestly pretty cool that these

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anti-differentiation problems are actually
doable, that we can write down the

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anti-derivatives.