Cogito
AP Calculus AB · Chapter 6 · Lesson 3
Substitution
The Chain Rule, run backwards.
12 problems · about 22 minutes · FUN-6.D
What this lesson teaches
The student evaluates integrals by substitution, adjusting limits where appropriate.
- Substitution reverses the Chain Rule.
- Take u as the inside of the composition and match du to the rest.
- For a definite integral, change the limits or convert back before evaluating.
Warm Up
Straightforward practice. Get the method working first.
5 problemsIf u = x⁴ + 3, then du = kx³ dx. What is k?
Answer 4
Why 4.
What must be done with the limits in a definite integral?
Answer Change them to u values, or convert back to x before evaluating.
Why Convert the limits or convert the answer.
If u = x³ + 2, then du = kx² dx. What is k?
Answer 3
Why Differentiate x³ + 2.
If u = 5x + 1, then du = k dx. What is k?
Answer 5
Why Differentiate 5x + 1.
Integral of u³ du gives u⁴ ÷ k. What is k?
Answer 4
Why Add one and divide.
Build It Up
The same ideas with more to keep track of.
3 problemsIntegral of u⁵ du gives u⁶ ÷ k. What is k?
Answer 6
Why Add one and divide.
Can a missing constant factor be compensated for? 1 yes, 0 no.
Answer 1
Why Constants can be moved.
Can a missing variable factor be compensated for? 1 yes, 0 no.
Answer 0
Why Variables cannot be moved past the integral sign.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the substitution process in order.
Answer 1. Choose u as the inside of the composition. 2. Differentiate to find du. 3. Rewrite the whole integral in terms of u. 4. Integrate, then convert back or change the limits.
Why du is found before the rewriting.
If u = x² − 7, then du = kx dx. What is k?
Answer 2
Why Differentiate x² − 7.
The Choice: For the integral of 2x(x² + 1)³, what should u be? Enter 1 for x² + 1, 2 for 2x.
Answer 1
Why u = x² + 1.
The Derivative: If u = x² + 1, then du = kx dx. What is k?
Answer 2
Why 2.