Integral

Integral step

So the integration by parts just tells us that the integral of uv prime is equal to uv minus the integral of

Let's see.

You've already dealt with vectors.

I'll call the plus the constant due to y.

Show integral

Welcome back.

That's equal to that.

That's the intuition behind the definite integral.

That'll come in useful later on.

Now, this is the part, and we've done this before, it's a

And the a's cancel out.

Almost every Laplace transform problem turns into an integration by parts problem.

So that's minus 1 over s e to the minus st, times v, sine of at, minus the integral.

So how do we evaluate this integral?

Where f of x,y is equal to P of x, y i plus Q of x, y j.

If that's the case, then what is v?

So let's do it traditionally.

So let's do x just to show you it really doesn't matter.

The bottom boundary is this surface.

And we have its derivative sitting right there.

And this is all going to be really good integration practice for us.

I'll keep the improper integral with us the whole time.

Well, now we have another hairy integral we need to solve.

Now, if we take the integral of both sides of this equation, we get uv is equal to the antiderivative of u prime v plus the antiderivative of uv prime.

But because this is a probability distribution, the area under the curve, that is the shaded area there, that area must integrate to 1.

And that's where the integral comes in.

Right.

This is part of the definition of the function.

This is essentially just this is the indefinite integral of x squared plus 1.

And now we can integrate both sides.

And now, we have to go back and find our integration by parts neuron.

And mine always disappears, so we have to reprove integration by parts.

I don't know what it is.

I said that also equals y.

I always forget it, but I just recorded it, so I do happen to remember it.

So your area for this function, or for this integral, is going to be 1.

let's make this what we want to solve for, so we can get the integral of uv prime is equal to we can just subtract this from that side of the equation, so it's equal I'm just swapping the sides, so I'm just solving for this, and to solve for this, I just subtract this from that, so it's equal to uv minus the integral of u prime v.

So what is our integral going to become with this substitution?

Well that's equal to what?

So we don't get an answer.

And I'll show it to you.

And I think this whole notion of these skills should be taught.

Welcome back.