### THE METHOD OF INTEGRATION BY POWER SUBSTITUTION

The following problems involve the method of power substitution. It is a method for finding antiderivatives of functions which contain th roots of or other expressions. Examples of such expressions are

and

.

The method of power substitution assumes that you are familiar with the method of ordinary u-substitution and the use of differential notation. Recall that if

,

then

.

For example, if

,

then

.

When using the method of power substitution, we will usually assume that

,

so that

.

For example, if

,

then

.

The goal of a power substitution will be to replace th roots of functions, which may not be easily integrated, with integer powers of functions, which are more easily integrated. For example, if we start with the expression

and let

,

then

and

.

Thus,

could be rewritten as

.

In general, to eliminate

,

let

.

Then

and

.

Recall the following well-known, basic indefinite integral formulas :

• 1.) , where
• 2.)
• 3.)
• 4.) , where is a constant
• 5.)
We can now apply the method of power substitution to an integral like

.

A COMMON WRONG ANSWER FOR THIS PROBLEM IS

.

Convince yourself that the answer is wrong by differentiating the right-hand side and showing it is NOT equal to the function on the left-hand side of the above equation. To properly integrate this problem, use the power substitution

so that

and

.

Now substitute into the original problem, replacing all forms of , getting

(Use polynomial division.)

.

Most of the following problems are average. A few are challenging. Make careful and precise use of the differential notation and and be careful when arithmetically and algebraically simplifying expressions. You will likely need to use the following rules for exponents.