Integral leading to the ln of a linear quantity

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Find

Do we have any integral formula with a quantity in the denominator?

Only

Can we use that when the denominator is ?

Not directly.

How else can we use it?

We can use it by doing a substitution which leads to this form.

Will work as a substitution?

No.

Why?

In order to get a form similar to we need to substitute for the whole denominator.

State that substitution.

We need to use

What do we do about dx?

We differentiate in order to express dx
in terms of du.

Do that.

From we get

We need to replace just dx. Solve for it.

Dividing by 3, we get

Combine these results to express our original problem in terms of u.

We get

How do we handle the ?

We can use the Scalar Multiple Rule for integration.

Do that.

Now we have

Do the integral.

This becomes

Are we done?

No.

What else must we do?

We want the result in term of x.

How do we get that?

We use the definition of our substitution, .

Do that.

We finally get

How can we check this result?

We can differentiate and compare with the original problem.

Do that.

Using the Chain Rule, we get

which checks.

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