Integration by parts and cyclic integrals (IB Math AA HL)

Key ideas

• The formula ∫u (dv/dx) dx = uv − ∫v (du/dx) dx is in the formula booklet.
• Pick u as the factor that simplifies when differentiated: ln x first, then powers of x, then eˣ or trig functions.
• For ∫ln x dx, write it as ∫1 × ln x dx with u = ln x.
• Some integrals need parts twice (x² eˣ). For eˣ sin x, the original integral comes back after two steps: call it I and solve for I.
• In a definite integral, evaluate the uv part between the limits as well as the remaining integral.

Worked example

Find the exact value of ∫₀¹ x e²ˣ dx.

1. Let u = x and dv/dx = e²ˣ, so du/dx = 1 and v = ½e²ˣ.
2. ∫₀¹ x e²ˣ dx = [½x e²ˣ]₀¹ − ∫₀¹ ½e²ˣ dx.
3. = ½e² − [¼e²ˣ]₀¹ = ½e² − ¼(e² − 1).
4. = ¼e² + ¼ = (e² + 1)/4.

Answer: (e² + 1)/4 ≈ 2.10

Exam-style HL AA questions on this topic

• Advanced Integration questions — 52 questions with mark schemes, plus free worked examples
• All HL AA Calculus questions by topic

Key formulas for integration by parts and cyclic integrals

Standard integrals (HL)

∫ 1/(√(a²−x²)) dx = arcsin x/a + C, ∫ 1/(a² + x²) dx = (1/a)arctan x/a + C

Integration by parts

∫ u dv = uv − ∫ v du

Volume of revolution (about x-axis)

V = π ∫ₐᵇ y² dx

Volume of revolution (about y-axis)

V = π ∫_cᵈ x² dy

In the notation of the IB formula booklet. All HL AA formulas · see them used in worked examples

Maths IA ideas that use integration by parts and cyclic integrals

• How big is my local lake? Estimating area with the trapezoidal rule
• How high did the lift go? Integrating a phone's accelerometer
• Trapezoidal rule or Simpson's rule? Areas under the normal curve
• Measuring inequality: fitting Lorenz curves and computing the Gini coefficient
• All HL AA IA ideas

Related in Calculus

• Maclaurin series and approximations
• Differential equations — separable and integrating factor
• Volumes of revolution and improper integrals

Revise Integration by parts and cyclic integrals for the exam

• HL AA revision notes — every unit summarised, with the key results to learn
• HL AA past papers — specimen papers and timed practice papers
• Formulas for Calculus — on the one-page HL AA formula sheet (free PDF), with the ones not in the IB booklet marked
• HL AA formula booklet by topic — every formula in the IB booklet's notation, with a free PDF
• IB Maths prediction papers — full papers in the IB format, with markschemes

Stretch yourself: original calculus and functions problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S133. All 24 →

Formulas in this topic: Integration by parts. All formulas A–Z