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[music] Now, we've seen how to
antidifferentiate some trig functions and

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polynomials.
Can we antidifferentiate, say, e to the x?

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So, the most general antiderivative of e
to the x, is just e to the x plus some

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constant C.
How do I know that?

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The derivative of e to the x is itself and
consequently the antiderivative of e to

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the x must also be itself and it's just
plus some single constant C.

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Since this is a function as to find in a
whole real line, well, great, we've found

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an antiderivative for e to the x.
What about an antiderivative for log?

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So, in symbols, right, what I'm looking
for is an antiderivative of the natural

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log of x, right, some function that
differentiates the natural log of x.

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But have we ever differentiated anything
and got log as the answer?

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Who knows?
But it turns out that the derivative of x

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log x is not so far off, right?
This is a derivative product, so by the

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Product Rule, it's the first thing times
the derivative of the second plus the

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derivative of the first thing times the
second thing, right?

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So, the derivative of x times log x is 1
plus log x, so not too far off.

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Now, you can kind of tweak this a little
bit, right?

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What's the derivative of x log x minus x?
Well, that's the derivative of x log x

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minus the derivative of x which is 1.
But I've just calculated the derivative of

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x log x.
It's 1 plus log x and then minus 1, what

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I'm left with then is just log x, right?
So, the derivative of x log x minus x is

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just log x.
So, in light of that, we can summarize the

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situation as follows.
So, yes, I mean, in, in light of this, I

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can say that the antiderivative of the
natural log is x times natural log of x

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minus x plus some constant C.
All of this is just so awful.

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It's just so depressing.
You know, the answers are coming, but

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they're coming out of the void.
I mean, they're just answers without any

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meaning or structure.
We're finding these antiderivatives just

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by really clever guessing, you know, and
it doesn't seem obvious how anyone would

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ever cook up these answers.
I mean, if, if you're not feeling really

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depressed right now, you probably should
be.

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You know, this is really bad.