Integration of a natural log function: Example 4

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We wish to find .

We need to express this in terms of a single variable, w.
What is a good candidate for replacement with this variable?

Let’s try to simplify the natural log function.

We also need to determine dw. Set up that up.

Determine dw.

Note that this combination appears in our problem.

Use these results to convert our problem to one expressed in terms of w.

We get

Can we integrate this directly?

No.

Why?

Because we don’t know of a function for which the derivative is ln(w).

Isn’t the integral equal to ?

No, because we would need a factor of
instead of just dw in the integral.

How can we solve this problem?

We can try Integration By Parts:

Set that up in general.

What should we take as u?

If we let u = ln(w), then we will get something simpler when we form du.

What should we take as dv?

As usual, dv is the rest of the integrand.
In this case, dv = dw.

Determine du.

.

Determine v.

.

Use these results in Integration by Parts.

Rearrange the first term and cancel in the second.

Do the integration.

Are we done?

No.

What do we do next?

We convert this result to an expression in x.

Do that.

We get

Let’s try the same problem by using our Table Of Integrals.
Which of the numbered equations in the table does
look like?

It is similar to #1, with an appropriate choice of “a”.

What should the value of “a” be?

We need a = 1.

Do we get the same result?

Yes.

Check this result by differentiation to see if we get our original integrand. Set up the differentiation.

Using the Product Rule in the first term, we get

Do the indicated derivatives.

We get

Cancel and simplify

We get

which checks.

The end. If you found this helpful and would recommend that I create more pages like this one, please let me know: Email to John Taylor

General Contents

Detailed Contents

Index