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?
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 arrow u = 2
x = 2 arrow u = 3
Here's the integral with the limits of integration fixed:
Now we can use the FTC to evaluate the integral.