Loading subject page…
- Calculus
- Techniques of Integration
- Integration by Parts: Definite Integrals
Micro-course: Techniques of Integration
4,672 views
Learn key concepts and earn your certificate.
Integration by Parts: Definite Integrals
00:00
01:23
- Understand how Definite Integrals in Integration by Parts extends the standard integration by parts formula to evaluate integrals over closed intervals with defined endpoints
- Identify when a definite integral requires integration by parts — especially when the integrand is a product of two functions with no straightforward antiderivative
- Apply the LIATE rule (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) to correctly assign which function to differentiate and which to integrate
- Analyze how the limits of integration must be adjusted when using substitution within a definite integral problem, such as setting t = 1 + x²
- Evaluate boundary terms by substituting the upper and lower limits directly into the product expression at each step of the solution
- Recognize how functions like the inverse tangent — which have no standard integral formula — can be rewritten as a product with the constant 1 to enable integration by parts
- Compare the roles of the natural logarithm and inverse trigonometric results that emerge as final answers in definite integration by parts problems
- Explore how this technique connects to related strategies such as trigonometric substitution, partial fractions, and improper integrals in higher-level calculus
Definite Integrals in Integration by Parts: What It Is and Why It Matters
Definite integrals in integration by parts is a technique used to evaluate the integral of a product of two functions over a specific interval — say, from x = 0 to x = 1. Rather than finding a general antiderivative and stopping there, you also apply the limits of integration directly, computing an exact numerical area under a curve. This is essential in AP Calculus BC, college Calculus II courses across the US, and in fields ranging from electrical engineering to economics.
The general formula is:
Integral from A to B of u dv = [u · v] evaluated from A to B − Integral from A to B of v du
The boundary term [u · v] from A to B means you substitute the upper limit and then subtract the result at the lower limit — just like you would with any definite integral using the Fundamental Theorem of Calculus.
How to Choose u and dv: The LIATE Strategy
Choosing the right assignment for u and dv is the most critical step. The LIATE rule gives a reliable priority order: Logarithmic functions first, then Inverse trigonometric, Algebraic (polynomials), Trigonometric, and Exponential last. The function higher on this list is typically assigned as u (the one to differentiate), while the remaining function becomes dv (the one to integrate).
For example, when evaluating the definite integral of the inverse tangent of x from 0 to 1, the function arctan(x) has no standard antiderivative in your formula sheet. The solution is to rewrite the integrand as arctan(x) · 1, assigning u = arctan(x) and dv = 1 dx. This is a classic move tested on AP Calculus BC free-response questions and college midterm exams nationwide.
Working Through the Inverse Tangent Example Step by Step
Once u and dv are assigned, the derivative du = 1/(1 + x²) dx and v = x are computed. Substituting into the formula gives:
[x · arctan(x)] from 0 to 1 − Integral from 0 to 1 of x/(1 + x²) dx
The first term is evaluated directly at the endpoints: at x = 1, the result is 1 · arctan(1) = π/4; at x = 0, the result is 0. So the boundary term contributes π/4.
The remaining integral — x/(1 + x²) — is solved using substitution. Setting t = 1 + x², the derivative dt = 2x dx, which means x dx = dt/2. The limits of integration shift: when x = 0, t = 1; when x = 1, t = 2. The integral becomes (1/2) · Integral from 1 to 2 of 1/t dt, which evaluates to (1/2) · ln(2). Since ln(1) = 0, the calculation is clean and direct.
The final answer is: π/4 − (1/2) · ln(2)
Real-World Applications and Exam Relevance
This technique appears in real US academic and professional contexts far beyond the classroom. In electrical engineering programs at MIT or Georgia Tech, integration by parts with definite integrals is used to compute signal energy over time intervals. In probability and statistics, it appears when deriving expected values for continuous distributions — relevant to students preparing for actuarial exams or AP Statistics extensions.
On the AP Calculus BC exam, definite integration by parts problems frequently appear in both the multiple-choice and free-response sections. Students in college Calculus II courses at state universities across the US encounter this in unit tests and final exams. Understanding how to adjust limits during substitution, evaluate boundary terms carefully, and recognize logarithmic outcomes are the three skills that separate students who score well from those who lose partial credit.
Mastering definite integrals in integration by parts also builds the foundation for tackling improper integrals, trigonometric integrals, and rational function integration — all of which appear in upper-division math and physics courses.
Frequently Asked Questions
Definite integrals in integration by parts is a calculus technique that evaluates the integral of a product of two functions over a specific numerical interval. Instead of finding just a general antiderivative, you also apply upper and lower limits of integration to compute the exact area under a curve. The formula adds a boundary term — [u · v] evaluated from A to B — alongside the remaining integral. This makes it ideal for functions like arctan(x) that have no simple antiderivative on their own. ---
Indefinite integration by parts produces a general antiderivative plus a constant C, valid for any value of x. Definite integration by parts goes one step further — the limits of integration are applied at each stage, including to the boundary term and any substitution used along the way. The result is a specific numerical answer representing the area under the curve between two fixed points, with no C needed. ---
Yes — this is a core topic in AP Calculus BC and appears regularly in both the multiple-choice and free-response sections. The College Board expects students to correctly assign u and dv, evaluate boundary terms at endpoints, and handle the resulting integral, sometimes requiring substitution as a second step. Practicing past AP free-response questions from the College Board website is one of the best ways to build confidence with this technique. ---
Absolutely. Definite integration by parts is a standard topic in Calculus II across virtually every US university — including community colleges, state schools, and Ivy League institutions. It typically appears in unit tests, midterm exams, and finals. Many professors include it alongside related techniques like trigonometric substitution, partial fractions, and improper integrals, so understanding it well gives you a strong foundation for the entire course. ---
Electrical engineers at companies like Lockheed Martin and Texas Instruments use this technique to calculate the energy of signals over defined time intervals. In biomedical research at institutions like Johns Hopkins, it helps compute expected values of probability distributions for patient data modeling. Actuarial science students use similar integrals when working with continuous probability density functions, making this skill relevant well beyond a single math class. ---
Not at all — if you're comfortable with basic derivatives, antiderivatives, and the Fundamental Theorem of Calculus, you have the prerequisites needed. The integration by parts formula itself is straightforward once you understand how to assign u and dv using a strategy like LIATE. Most students in AP Calculus BC or college Calculus II pick up the method quickly with a few guided examples and some practice problems. ---
The key signal for integration by parts is a product of two different types of functions — like a polynomial multiplied by a logarithm, or an inverse trig function times a constant. Trigonometric substitution is better suited when the integrand contains expressions like the square root of (a² − x²). Partial fractions work best for rational functions where the denominator can be factored. Learning to recognize these patterns is one of the most valuable skills you can develop in Calculus II. ---
Once you're confident with this technique, the natural next steps are trigonometric integrals, trigonometric substitution, and integration of rational functions using partial fractions. These topics build directly on your ability to recognize integrand structure and choose the right technique — skills sharpened through integration by parts practice. Improper integrals, which extend the definite integral concept to infinite limits, are another important topic that follows logically in most Calculus II curricula.
Introduction to Differential Equations explores a complementary area of Calculus, while Integration by Parts: Definite Integrals focuses on the specific concept covered in this video. Understanding both helps you build a stronger foundation in Calculus.
A basic understanding of Integration by Parts: Indefinite Integrals is helpful before diving into Integration by Parts: Definite Integrals. If you are starting from scratch, the Techniques of Integration series builds knowledge progressively, so beginning from the first video requires no prior background in Calculus.
The ideas in Integration by Parts: Definite Integrals show up in everyday Calculus contexts, often alongside concepts like Curves Defined by Parametric Equations. This video connects the theory to practical situations you may encounter in coursework or exams.
After Integration by Parts: Definite Integrals, the natural next step is Integration by Parts: Problem Solving in the Techniques of Integration series. Following the playlist in order helps concepts build on each other without gaps.
The Integration by Parts: Definite Integrals video runs for 1 minute, so you can cover the core concept in a single focused study session without needing a long block of time.
Related Micro-courses
6 Concepts
16 Concepts