Contents of this Calculus 2 episode:

Integration by parts, Integrate the product by parts, Partial integration, The casting, Integration by parts step by step, Cases with reverse casting, Reduction formula.

Partial integration was developed for integrating products.

The name comes from the fact that we will integrate the product in parts.

Let's see how.

We need to do some casting.

One factor of the product will be f, and the other one will be g'.

But it is not recommended to decide by tossing a coin.

So, it would be useful to have an idea how to pick the roles.

Let's try it this way first:

The purpose of partial integration is to turn a complicated integral into a simpler one.

The casting is good, when we achieve this goal.

This time we failed.

This integral is even more complicated than the original, because instead of x, it contains x2 .

What did we mess up?

Well, the thing is, there are two open roles, f and g'.

During the procedure, we differentiate f, integrate g', and these will appear in the new integral.

If we assign the role of g' to x, then when we integrate, we will increase x's exponent.

If we assign the role of f to x, then we will differentiate it, and x's exponent will be decreased.

We want to decrease the exponent.

Thus, we have to reverse the roles.

With this casting, we differentiate x, and it becomes 1.

For ex, it doesn't matter, it becomes ex, just like before.

Here, we achieved our goal this second integral is indeed simpler.

Let's see one more:

Based on the previous discussions, it seems that f=x2 will be the winning choice.

We achieved our goal: successfully turned a complicated integral into a simpler one.

But it is still not simple enough, so we do another partial integration.

We learned so far that it seems best to assign the role of f to the power of x in the product.

Let's make a note of this for ourselves.

CASTING:

But, life would be too easy without some annoying exceptions.

Here is this one, for instance:

Based on our notes, we should assign the roles like this:

But unfortunately, it leads to nothing. If you don't believe it, you can try and see yourself.

So, we reverse the roles.

We extend our list.

Cases where we should use reverse casting:

And there is one more thing.

Here are these functions from cases where reverse casting is necessary.

Well, let's try to integrate them.

We will need a bit of a trick.

We will use partial integration. Based on our list, we will call these f, and here comes the trick: g’=1

Let's see another one, too.

The purpose of partial integration is to turn a complicated integral into a simpler one.

In the previous slideshow we saw how partial integration works, and now we will solve a few splendid problems.

Here comes this one, for instance:

¶=f · g- f f’·g

Let's see one more:

f f·g’ =f·g- f f’·g

For partial integration, we assign the roles:

=f · g- f f’·g

Here comes another one.

Well, there is such a thing as:

¶

We should be careful with this partial integration thing, as it could be harmful in large doses.

But perhaps one more is safe.

Well, there is this thing, too:

And this one, too:

=f · g- f f’·g

The point of partial integration is that it turns complicated integrals into simpler ones.

The process that transforms a more complicated integral into a simpler integral is called reduction.

Here come a few reduction formulae. There is really no point in memorizing these, as we can calculate them at any time.

We can use these to finish off a few of the nastier integrals.

Calculus 2 episode