One way, if you did want to memorize it, you said, OK, the integration by parts says if I take the integral of the derivative of one thing and then just a regular function of another, it equals the two functions times each other, minus the integral of the reverse.

And so to actually calculate this, what I do is I take the cumulative distribution function of this point and I subtract from that the cumulative distribution function of that point.

So what do we want to make our v prime?

Cumulative 0 calculates the density function cumulative 1 calculates the distribution.

So what I did is I evaluated the cumulative distribution function at one to be right there.

If f prime of t is equal to cosine of a t, what is a potential f of t?

So the important topics to take away from this segment.

If we take out the constants from inside of the integral, we get e to the minus sc times f of c times the integral from 0 to infinity of f of t minus c dt.

So this is derivative.

Well, we can do a little pattern matching here, right?

When I introduced you to the unit step function, I said, you know, this type of function, it's more exotic and a little unusual relative to what you've seen in just a traditional Calculus course, what you've seen in maybe your

So when you take a derivative, when you take a partial with respect to just x, a pure function of just y would get

It's always good to use the exponential function, because that's easy to take the antiderivative of.

Well that's equal to what?

a function has been defined called the cumuluative distribution function that is a useful tool for figuring out this area.

And we're going to see some concrete examples of taking a

So how do we evaluate this integral?

Laplace Transform.

K 0 calculates the density function K 1 calculates the distribution.

I'm going to tell you that if I were to take the integral of this function from minus infinity to infinity, so essentially over the entire real number line, if I take the integral of this function, I'm defining it to be equal to 1.

So what the cumulative distribution function is essentially let me call it the cumulative distribution function it's a function of x.

And so by the same intuitive argument, you could say that the limit from minus infinity to infinity of our Dirac delta function of t dt is also going to be 1.

And just by the definition of how the properties of integrals work, we know that we can split this up into two integrals, right?

Then that equals the derivative of the first times the second function, plus the first function times the derivative of the second.

Let's implement this in our code.

If the integral of the sum of two functions is equal to the sum of their integrals.

Burning lots of midnight oil.

So what would be the volume of that sliver?

So if we take from zero to infinity, what I'm saying is taking this integral is equivalent to taking this integral.

little bit, let me get out of pen tool so the cumulative distribution function is right over here then you say true when you make that Excel call.

And then we started with the Laplace Transform of y, and then you can almost view the Laplace Transform as a kind of integral, so we kind of take the derivatives, so then you get y.

If it's not, you might want to review the definite integration videos.

So plus f1 of x, that's just the first function, times the derivative of the second function.

And then, well, you would end up saying that the antiderivative is just the natural log of u.

Your plane starts loading in just a few minutes now.

And to solve for psi, we just say, OK, the partial derivative of psi with respect to x is going to be this thing.

You can equally define over states in action pairs.

So it's kind of borrowing that notation, because this function of s is kind of an integral of y of t.

So if this was a 1 over an x, or 1 over u, it's just the natural log of it.

What is a differential equation?

Well let's take the antiderivative of both sides or let's take the integral of both sides.

Our JavaScript interpreter is going to have to compute the value of 6 factorial, a function call within a function call within a function call.

Integral

Laplace transform of the derivative of a function, and the Laplace transform of the function.

Generally speaking, we can compute the comoving distance, distance in coordinates that do expand with the expanding space.