AP Calculus BC
6.3 Antidifferentiation by Parts
Objective: able to use integration by parts to evaluate definite and indefinite integrals; to use
tabular integration or solving for the unknown integral in order to evaluate integrals that require
repeated integration by parts.
Product Rule in Integral Form
When u and v are differentiable functions of x,
Integration by Parts Formula
∫ u dv=
uv – ∫v du
Side note, p. 341. LIPET
1. Evaluate
∫ x cos x dx using integration by parts.
2. Evaluate
∫ xex dx using integration by parts.
3. Evaluate
∫ 2x2ex dx by repeated use of integration by parts.
Integrals of the form ∫f(x) g(x) dx, in which f can be differentiated repeatedly to become zero and g can be
integrated repeatedly without difficulty, are natural candidates for integration by parts. If many repetitions are
required, tabular integration can organize the calculation and save time.
4. Evaluate
∫ 2x2ex dx by using tabular integration.
5. Solve the initial value problem
=
+ 2 sin ℎ
= 0
= 2.
Rate yourself on how well you understood this lesson.
I don’t get
it at all
I sort of
get it
I understand
most of it but I
need more practice
I understand
it pretty well
I got it!
1
2
3
4
5
What do you still need to work on?