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
Evaluate the definite integral using Way 1(first integrate the indefinite integral, then use the FTC).
Answer
Using substitution, we find that
We use the FTC:
Example 6
Evaluate the definite integral using Way 1(first integrate the indefinite integral, then use the FTC).
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.