Burning lots of midnight oil.

Now, chemically speaking, this is equivalent to lowering the volume of a reaction vessel or a reaction mixture.

Integral

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

Integral step

Let's see.

You've already dealt with vectors.

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

When you evaporate seawater, the first thing to crystallize out is calcium carbonate.

We say there were nuclear weapons on the plane when it went down.

Ordnance loading, weapons strip and dropship prep details will have seven hours.

Show integral

Welcome back.

That's equal to that.

That's the intuition behind the definite integral.

The ship was stowed with arms.

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 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.

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

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

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.

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

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

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.

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

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

And now we can integrate both sides.

And that's where the integral comes in.

Right.

This is part of the definition of the function.

It's on the tip of the spear.

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.