Integration by Parts for AP Calculus BC | Tips & Strategies

Always choose u from the highest priority in LIATE. For example, in ∫x e^x dx, you would choose u = x and dv = e^x dx.

Step-by-Step Example

Let’s solve: ∫x e^x dx

1. Choose u = x, dv = e^x dx
2. Then du = dx, v = e^x
3. Apply formula: ∫u dv = uv – ∫v du
4. ∫x e^x dx = x e^x – ∫e^x dx
5. = x e^x – e^x + C

Common Mistakes to Avoid

• Choosing u incorrectly: If you pick the wrong function for u, the problem becomes harder instead of easier.
• Forgetting the minus sign: Always double-check signs when subtracting the ∫v du term.
• Not simplifying: Many Integration by Parts problems simplify dramatically—don’t stop too early.

Integration by Parts with Repeated Use

Sometimes, you’ll need to apply Integration by Parts more than once. For example:

∫x² e^x dx

• First, let u = x², dv = e^x dx.
• After one round, you’ll still have another polynomial term to integrate.
• Reapply Integration by Parts until the polynomial reduces.

Practice Problems (Try These)

1. ∫x sin(x) dx
2. ∫ln(x) dx
3. ∫x² e^x dx
4. ∫arctan(x) dx

(Detailed step-by-step solutions available on RevisionDojo).

Why Integration by Parts Matters on the AP Exam

• It appears frequently in both multiple-choice and free-response questions (FRQs).
• It often combines with other methods like u-substitution or trigonometric identities.
• Mastery of this technique shows flexibility in problem solving, which AP graders reward.

Study Tips with RevisionDojo

• Use the LIATE rule as a quick decision-making tool.
• Practice multiple examples daily to recognize patterns.
• Review solved examples and FRQs on RevisionDojo, where step-by-step AP-level practice is available.

Frequently Asked Questions

Q: Do I always need to use LIATE to choose u?
A: Yes, LIATE is the most reliable method, but with practice you’ll develop intuition for choosing u.

Q: Can Integration by Parts combine with substitution?
A: Absolutely. Many AP exam questions require combining techniques for full solutions.

Q: Is Integration by Parts tested more in AP Calculus AB or BC?
A: It is primarily emphasized in AP Calculus BC, though AB students may occasionally see it.

Q: Where can I get more step-by-step practice?
A: RevisionDojo provides AP Calculus study guides, formula sheets, and problem sets designed for exam success.