Contents of this Calculus 2 episode:

Summary problems for whole indefinite integration, Integration by substitution, useful substitutions, dx/du, integration by parts, Linear expression under the radical, Non-linear expression under the radical, Integration by parts, Composite function, The S4 integration rule, Special cases.

And now, let’s see how to solve an integration problem.

Well, we have to ask ourselves a few questions.

Is there a thing like this in the integral:

If yes, then there are two possible cases.

There is a linear expression under the radical

There is a non-linear expression under the radical

In this case it is worth to try substitution:

In this case, it is usually a good idea to convert the radical thingy:

and then we can use S2

Is there or in the integral?

x is raised to the first power

x is raised to some power

This is presumably an S2

And this one is partial integration.

If SOMETHING is linear, then we should use partial integration.

If SOMETHING is not linear, then we need the S4 formula:

And now we can get to the problems.

We have to ask ourselves these questions before solving each of these problems,

so that we could pick the appropriate formula.

Here is this one, for instance.

It contains a radical. Let’s see what we listed for radical cases like this:

It contains a logarithm. Let’s see what we listed for logarithmic cases like this:

It contains ex. Let’s see what we listed for ex cases like this:

And now let’s see what the difference is between the two problems.

At first sight, not much.

But let’s just see what we listed for ex cases like this:

Finally, we have these three.

There are radicals in all, but each must be solved in a different way.

Let’s see our notes:

Theoretically, this first one should be solved by substitution, but actually it is very simple.

Well, the second one is really for substitution, due to this x here.

Calculus 2 episode