Cogito
AP Calculus BC · Chapter 6 · Lesson 1
Integration by Parts
The Product Rule, run backwards.
12 problems · about 22 minutes · FUN-6.E
What this lesson teaches
The student evaluates integrals using integration by parts, including repeated application.
- Integration by parts is the Product Rule integrated.
- The integral of u dv equals uv minus the integral of v du.
- Choose u to be the factor that simplifies when differentiated.
Warm Up
Straightforward practice. Get the method working first.
5 problemsIntegrating x⁵ times eˣ. How many applications of the formula?
Answer 5
Why 5.
How should u be chosen?
Answer The factor that simplifies when differentiated.
Why The one that gets simpler.
Integrating x² times eˣ. How many applications?
Answer 2
Why One per degree.
Integrating x times sin x. Which should be u? 1 x, 2 sin x.
Answer 1
Why The algebraic factor.
Integrating ln x. Which should be u? 1 ln x, 2 dx.
Answer 1
Why Logarithms come first in the ordering.
Build It Up
The same ideas with more to keep track of.
3 problemsIf u = x then du = k dx. What is k?
Answer 1
Why The derivative of x.
If dv = eˣ dx then v = k times eˣ. What is k?
Answer 1
Why The integral of eˣ.
Integrating x⁴ times eˣ. How many applications?
Answer 4
Why One per degree.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the integration by parts process in order.
Answer 1. Choose u as the factor that simplifies when differentiated. 2. Set dv to the rest and find v and du. 3. Apply the formula. 4. Evaluate the new integral, repeating if needed.
Why The choice of u comes first.
Integrating x times cos x. Which should be u? 1 x, 2 cos x.
Answer 1
Why The algebraic factor.
The Choice: Integrating x times eˣ. Which should be u? 1 x, 2 eˣ.
Answer 1
Why u = x.
The Repeats: Integrating x³ times eˣ. How many applications of the formula are needed?
Answer 3
Why 3.