Introduction
Integration by Parts is one of the most important integration techniques tested on the AP Calculus BC exam. While many students feel confident with u-substitution, Integration by Parts requires a different type of problem-solving skill. Understanding when and how to apply the method is key to solving advanced integrals, especially those involving products of functions like logarithms, exponentials, and trigonometric expressions.
In this guide, we’ll break down the Integration by Parts formula, show you step-by-step examples, highlight common mistakes, and provide study tips to ensure you’re ready for the AP Calculus BC exam.
What is Integration by Parts?
Integration by Parts is based on the product rule for differentiation but applied in reverse. It’s especially useful when the integral involves a product of two functions that can’t be solved with basic substitution.
The formula is:
∫u dv = uv – ∫v du
Where:
- u = the function you choose to differentiate
- dv = the function you choose to integrate
- du = derivative of u
- v = integral of dv
When to Use Integration by Parts
Students often ask, “How do I know when to use Integration by Parts?”
You should consider it when the integral involves:
- A product of polynomial and exponential functions (e.g., ∫x e^x dx)
- A product of polynomial and trigonometric functions (e.g., ∫x sin(x) dx)
- Logarithmic or inverse trig functions (e.g., ∫ln(x) dx, ∫arctan(x) dx)
The LIATE Rule for Choosing u
Choosing u correctly is the most important part of Integration by Parts. The LIATE rule helps:
- L: Logarithmic (ln, log)
- I: Inverse trig (arctan, arcsin, etc.)
- A: Algebraic (polynomials like x^2, x^3)
- T: Trigonometric (sin, cos, tan)
- E: Exponential (e^x, a^x)
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
- Choose u = x, dv = e^x dx
- Then du = dx, v = e^x
- Apply formula: ∫u dv = uv – ∫v du
- ∫x e^x dx = x e^x – ∫e^x dx
- = 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)
- ∫x sin(x) dx
- ∫ln(x) dx
- ∫x² e^x dx
- ∫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.
