[music] With all that we've learned so far, it's super easy to differentiate sine
squared x, so I'm going to differentiate sine squared.
Well, it's the chain rule, right? It's the derivative of the outside
function, which is the squaring function. So, it's 2 times the inside function,
which is sine x, times the derivative of the inside, which is cosine x.
So there, that's the derivative of sine squared x.
What if I wanted to go backwards? What if I wanted to anti-differentiate
sine squared x? I want to anti-differentiate sine squared
x. I'm looking for a function whose
derivative is sine squared x. Well, me and my first guess for central
function might be sine cubed over 3. Because if I differentiate this while I'm
differentiating the outside function, which is cubing, so it's 3 times the
inside function squared divided by 3, and that's looking good.
There, I've got a sine squared x. The problem is that, now I've got to
multiply this by the derivative of the inside function, which is sine, and the
derivative of sine is cosine. So, the derivative of this is not sine
squared x, it's sine squared x times this extra cosine.
Oh, it seems really hard. Well, there's a so-called half angle
identity. This is the trick.
That sine squared x is, according to half angle identity, 1 minus cosine of 2x all
over 2. Well, that's true.
But how does that identity help? Well, since sine squared x is 1 minus
cosine 2x all over 2, that means the anti-derivative for sine squared x is
equal to the anti-derivative of 1 minus cosine 2x over 2.
Now to do this, this is the anti-derivative of 1 half times something.
So, I can pull out that factor of 1 half and just figure out how to
anti-differentiate 1 minus cosine of 2x. Now, this is an anti-derivative of a
difference, which is the difference of anti-derivatives.
So, I want to anti-differentiate 1 and I want to anti-differentiate cosine 2x.
Now, what's an anti-derivative of 1? Well, it's just x.
And what's an anti-derivative of cosine 2x?
Well, one such anti-derivative is sine of 2x.
But then, I've got to divide by 2 to compensate for the chain rule here.
It'll add plus c. So, there we go.
I could simplify this a little bit. I could write this as x over 2 minus sine
2x over 4 plus c, and there is my general anti-derivative for sine x.
Maybe you really like this sort of thing. Well, if you are really having fun with
this kind of thing, here's a little challenge that you can play with.
You can try to find an anti-derivative say, for sine to the 4th, right?
And if you don't like the number 4, you can make the 4 some other number, right?
But it's pretty fun. Then, you can try to apply some of the
trig identities to replace sine to some power with things that you can currently
anti-differentiate. It's honestly pretty cool that these
anti-differentiation problems are actually doable, that we can write down the
anti-derivatives.