Integral of a linear expression to a fractional power

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Find

Do we have any integral formula with a power?

Only

Can we use this when the power is a fraction?

Yes.

How do we handle the ?

We can do a substitution which leads to the power form.

Will work as a substitution?

No.

Why?

We need to substitute a new variable for the quantity
raised to the power:

State that substitution.

We need to use

What do we do about dx?

In order to express dx in terms of du,
we differentiate

Do that.

From we get

We need to replace just dx. Solve for it.

Dividing by 8, we get

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

We get

How do we handle the ?

We can simplify and 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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