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.

Show integral

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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 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.

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?

This is equal to the integral from 0 to infinity of e to the minus st, times f prime of t, dt.

It makes sense, because normally when we're doing antiderivatives, you just take you know, when you

Now this transformation has doubled the area of the original figure, and therefore we have just learned that the area of the triangle equals the area of this rectangle divided by two.

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

So if we have this unit step function, this thing is going to zero out this entire integral before we get to c.