Programmed tutorial: Integration of a natural log function: Example 3

No.

Why?

Because we don’t know of a function for which the derivative is x²*ln(7x).

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(7x), 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 = x²*dx.

Determine du.

Determine v.

Use these results in Integration by Parts.

Rearrange the first term and cancel in the second.

Do the integration.

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 #3, with an appropriate choice of “a”.

What should the value of “a” be?

We need a = 7.

Do we get the same result?

Yes.

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

Using the Product Rule in the first term, we get

Do the indicated derivatives.

We get

Cancel and simplify.

We get , which checks.

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