What did Laurie do wrong? What is the correct value of the integral?

Answer

When finding the indefinite integral, Laurie forgot to put back the original variable. She should have written

When we use the FTC with the correct antiderivative, -ln|4 – x|, we get

The moral of Laurie’s story is to keep the two steps separate when evaluating a definite integral using Way 1. First, evaluate the indefinite integral. Remember to put the original variable back in. Then, after you’re all done with the indefinite integral, use the FTC. Keep your calculations for the two steps separate so you don’t get mixed up!

Example 2

Evaluate the definite integral using Way 1(first integrate the indefinite integral, then use the FTC).

Answer

First we work out the indefinite integral. We have

So

Now we use the FTC with the antiderivative

Here goes:

Example 3

Evaluate the definite integral using Way 1(first integrate the indefinite integral, then use the FTC).

Hint

Rewrite the integrand.

Answer

Following the hint, we can rewrite the integrand:

This is a little easier to work with. First we evaluate the indefinite integral using substitution. Take

u = 3x + 4, so u’ = 3. Then

Now we use the FTC with the antiderivative .

Example 4

Evaluate the definite integral using Way 1(first integrate the indefinite integral, then use the FTC).

Answer

Using substitution, we find that

Now we use the FTC.

Example 5

Answer

Using substitution, we find that

We use the FTC:

Example 6

Answer

First we figure out the indefinite integral. Take u = ln x. Then . So

Using the FTC, we get

Example 7

A test contained the following question:

KT wrote down the following answer:

What did KT do wrong? What is the correct value of the integral?

Answer

KT took u = x + 1, so du = dx.

She neglected to change the limits of integration when doing the substitution. The limits 1 and 2 are values of x, not values of u. She wrote this:

Which means this:

However, KT continued as though it meant this, which is not true:

To find the correct value of the integral, we need to finish the substitution. Since u = x + 1, we have

x = 1 → u = 2

x = 2 → u = 3

Here’s the integral with the limits of integration fixed:

Now we can use the FTC to evaluate the integral.

We conclude

Example 8

Evaluate the definite integral by substitution, using Way 2.

Answer

First we need to do the substitution. Take

Then

So

Now we can use the FTC on this simpler integral.

That’s an exact answer, so we leave it like that. We conclude

Example 9

Evaluate the definite integral by substitution, using Way 2.

Answer

Take

u = -x

du = (-1)dx

Then

We still need to change the limits of integration. Since u = -x, this part is pretty simple.

So

Now we can use the FTC. The function sec2u is the derivative of tan u, so

We conclude

Example 10

Evaluate the definite integral by substitution, using Way 2.

Answer

First we do the substitution. Take

u = (x3 – 3x + 2)

du = (3×2 – 3)dx

= 3(x2 – 1)dx

Then

To change the limits of integration we have to do a little work this time. When x = -2, we have

u = (-2)3 – 3(-2) + 2 = 0.

When x = 0 we have

u = (0)3 – 3(0) + 2 = 2.

We fix the limits of integration:

Now we use the FTC:

Example 11

Evaluate the definite integral by substitution, using Way 2.

Answer

We do the substitution first, remembering that this includes changing the limits of integration. Take

We introduce a factor of 4 to the integrand in order to start the substitution.

We still need to change the limits of integration. When we have

and when x = π we have

After we change the limits of integration, the substitution is complete:

Now we can use the FTC to evaluate the integral.

We conclude

Example 12

Evaluate the definite integral by substitution, using Way 2.