Integrating Using Integration by Parts
Reverse the product rule with a justified parts choice, compatible bounds, and a derivative check.
College Board Topic 6.11 · CED p. 123 (PDF p. 128) · CED effective Fall 2020; project course outline 2026–27.
BC extension: keep the shared AB accumulation, algebra, and substitution lessons active. Integration by parts reverses the product rule: ∫u dv = uv−∫v du. Choose u so its derivative gets simpler, and choose dv so you can integrate it. This is often useful for x eˣ, x sin x, or ln x (where dv=dx). It is not the same as multiplying two separate integrals.
For a definite integral, use [uv]ₐᵇ−∫ₐᵇ v du with compatible variables and bounds. You may need the method more than once. Differentiate an indefinite result to verify it, and retain the interval where all terms are defined.
Let's work through it
- For ∫x eˣdx, choose u=x, dv=eˣdx, giving du=dx and v=eˣ. Reason: Differentiating x simplifies it, and eˣ is easy to integrate.
- Apply parts: x eˣ−∫eˣdx=(x−1)eˣ+C. Reason: The product-rule rearrangement subtracts the remaining integral.
- Differentiate (x−1)eˣ to get x eˣ. Reason: Verification confirms the subtraction and coefficient.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
For ∫ln x dx, what can dv be?
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
For ∫ln x dx, what can dv be?
Worked answer and reason
dx, so v=x.
Treat ln x as ln x·1.
Retrieval 2
What is the sign in the parts formula?
Worked answer and reason
uv minus ∫v du.
Rearrange d(uv)=u dv+v du.
Catch the tempting mistake
- ∫u v equals (∫u)(∫v).
- Integration does not distribute over products. Parts includes a correction integral derived from the product rule.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
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+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 6 problem 21 · No calculator · analytic
- Unit 6 problem 22 · No calculator · analytic
- Unit 6 problem 23 · No calculator · analytic
For the teacher
- Begin with the product rule derivation, then have students defend their choice of u. Include a definite-bound example and a derivative check.
Glossary and deeper reading · 3 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Product rule
The product rule differentiates a product of two differentiable functions. It gives (fg)′=f′g+fg′, changing one factor at a time and adding the results. For example, the derivative of x sin x is sin x+x cos x. The product of the two derivatives is generally not the derivative of the product.
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Derivative
A derivative gives the instantaneous rate of change of a function with respect to its input. At a point, it is the finite limit of the difference quotient when that limit exists. For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6. A derivative function collects those rates across the inputs where the function is differentiable.
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Definite integral
A definite integral accumulates a function over a specified interval as a limit of Riemann sums when that limit exists. Its sign includes the function’s sign and the orientation of the limits. For example, the integral of the constant −2 from 0 to 3 is −6, although the geometric area is 6. In a rate model, the result has rate units multiplied by input units.