Integration by Parts (BC Only) — AP Calculus BC
1. Core Integration by Parts Formula ★★☆☆☆ ⏱ 3 min
Integration by parts reverses the product rule for differentiation, and is used to integrate products of distinct function types that cannot be solved with u-substitution alone. It splits a complex original integral into a simpler product term and an easier-to-evaluate new integral.
Exam tip: When integrating a single function like $\ln x$ or $\arctan x$ (not an explicit product), rewrite it as $f(x) \cdot 1 dx$, set $u = f(x)$ and $dv = 1 dx$ to convert it to a standard problem.
2. Repeated and Cyclic Integration by Parts ★★★☆☆ ⏱ 4 min
Most integrals involving higher-degree polynomials, or products of exponentials and trigonometric functions, require more than one application of integration by parts. For an $n$-th degree polynomial $u$, each application reduces the degree of $u$ by 1, so $n$ applications are needed to reduce $u$ to a constant.
A BC-exclusive special case is cyclic integration by parts, which occurs when integrating products of the form $e^{ax} \sin(bx)$ or $e^{ax} \cos(bx)$. After two applications, the original integral reappears on the right-hand side, allowing you to solve for it algebraically. This is a very common AP exam problem.
Exam tip: Always add the constant of integration $+C$ only after you have isolated the original integral $I$ on the left-hand side. Adding $C$ early will lead to an incorrect constant multiple in your final result.
3. Tabular Integration (DI Method) ★★★☆☆ ⏱ 3 min
Tabular integration is a time-saving shortcut for repeated integration by parts that works exclusively when one function (the $u$ term) differentiates to zero after a finite number of steps, which is almost always a polynomial of any degree. The method organizes work into two columns: $D$ (differentiate $u$ repeatedly) and $I$ (integrate $dv$ repeatedly). Stop when $D$ reaches zero, then sum alternating-sign diagonal products to get the antiderivative.
Exam tip: Never use tabular integration for cyclic cases like $e^x \sin x$. You will never reach a zero derivative, so the method will not work and will lead to an incorrect result.
4. AP Style Practice Problems ★★★★☆ ⏱ 4 min
Common Pitfalls
Why: Students mix up LIATE order, needing to integrate $\ln x$ for $v$ which unnecessarily complicates the problem
Why: Students forget the original integral reappears and must be solved for algebraically
Why: Students focus on integrating the $\int v du$ term and forget the boundary term entirely
Why: Students confuse the sign flip from the $- \int v du$ term in the core formula
Why: Students incorrectly assume integration by parts only works for explicit products