The purpose of this set of supplemental notes is to provide you detailed examples of how to use this method to integrate complicated functions. If you think a little bit about this method, you will begin to realize that it is the reverse process of the chain rule. In fact, it is based on the chain rule, and you choose u based on the same criteria as you do in the chain rule. In the first few examples we will only work with indefinite integrals, then we will tackle some definite integrals.

The integrals that we will be considering are of the form ò f (g (x)) g' (x) dx. Here are the steps you must perform when using integration by substitution.

1.

Let u = g (x).

2.

Find du/dx = g' (x).

3.

Let du = g' (x) dx. Now, make sure that this is contained in the original integral. If not, then you cannot use this method.

4.

Substitute u in for g (x) and du in for g' (x) dx.

5.

Integrate the new integral.

6.

Substitute g (x) in for u and add + c to the answer.

EXAMPLE 1:

SOLUTION:

EXAMPLE 2:

SOLUTION:

Notice that we did not have 3dx, so we had to solve for dx in terms of du. Multiplying by a constant is acceptable, but if du = 3x dx, then we could not integrate it. The reason for this is that we cannot multiply by a variable because it will change the original integration problem.

EXAMPLE 3:

SOLUTION:

Again, we had to solve for dx in terms of du.

EXAMPLE 4:

SOLUTION:

Here we did not have to solve for dx in terms of du. It was a straight substitution.

EXAMPLE 5:

SOLUTION:

Notice again that we had everything in the original to complete the substitution except for the constant. Again we had to solve for dx in terms of du.

EXAMPLE 6:

SOLUTION:

EXAMPLE 7:

SOLUTION:

EXAMPLE 8:

SOLUTION:

EXAMPLE 9:

SOLUTION:

Let u = tan x, then du = sec ² x dx.

EXAMPLE 10:

Lets do the integral in example 9 again, this time we will make the following u substitution.

Let u = sec x, then du = sec x tan x dx.

SOLUTION:

It is the exact same problem, but we got different answers. What is going on here? They are both correct solutions to this integration problem. In fact the second answer only differs by a constant. The constant comes from using the trig identity tan ² x + 1 = sec ² x.

The steps for doing integration by substitution for definite integrals are the same as the steps for integration by substitution for indefinite integrals except we must change the bounds of integration and we do not need to sub back in for u.

1.

Let u = g (x).

2.

Find du/dx = g' (x)

3.

Let du = g' (x) dx. Now, make sure that this is contained in the original integral. If not, then you cannot use this method.

4.

Substitute u in for g (x) and du in for g' (x) dx.

5.

Find the new bounds of integration by plugging in the lower bound into u. That result will be the new lower bound. Then plug in the upper bound into u. This will be the new upper bound.

6.

Integrate the new integral.

7.

Plug in the new bounds and evaluate.

EXAMPLE 11:

SOLUTION:

EXAMPLE 12:

SOLUTION:

EXAMPLE 13:

SOLUTION:

EXAMPLE 14:

SOLUTION:

Why does the integral evaluate to 0? Remember that the definite integral is used to find the area between two curves. If the upper and lower bounds are the same, there will be no area between the curves. Therefore, the area will be zero. So remember that when the upper and lower bounds are the same, then the answer to the integral is 0.