SL 5.5 — Integration (Anti-differentiation & Definite Integrals)

Integration is the inverse process of differentiation and is used to reconstruct a function from its rate of change.
In IB Mathematics, integration is applied to find antiderivatives, determine functions using boundary conditions,
and calculate accumulated quantities such as area, displacement, and total change.

Concept IB Definition & Role
Indefinite integral An indefinite integral represents the family of all antiderivatives of a function and always includes an arbitrary constant.
It is written as ∫ f(x) dx = F(x) + C.
Constant of integration (C) The constant C appears because differentiation removes constant values.
Different values of C represent parallel curves with the same rate of change.
Definite integral A definite integral represents the signed accumulation of a function between two limits.
When the function is positive, it corresponds to area under the curve.
Fundamental Theorem of Calculus This theorem links differentiation and integration by stating that the definite integral can be evaluated using an antiderivative.

📌 1. Antiderivatives and indefinite integrals

  • An antiderivative of a function f(x) is a function F(x) such that F′(x) = f(x).
  • Indefinite integrals represent an infinite family of functions rather than a single curve.
  • The constant of integration must always be included unless additional information is provided.
  • Failing to include +C is considered a conceptual error in IB examinations.
  • Indefinite integrals are primarily used to reconstruct equations from given rates of change.

Power rule for integration

  • The power rule applies to all polynomial terms where the exponent is not −1.
  • The exponent increases by one during integration.
  • The coefficient is divided by the new exponent.
  • The constant of integration must be appended at the end.
  • This rule forms the foundation of almost all basic IB integration problems.

🧠 Examiner Tip

  • Always write +C unless the question explicitly states “find the particular solution”.
  • Marks are often awarded for method even if arithmetic errors occur later.
  • Using differentiation to check your answer strengthens accuracy and confidence.

📌 2. Using boundary conditions

  • Boundary conditions allow a specific function to be determined from an indefinite integral.
  • A boundary condition provides a known value of the function at a specific input.
  • Substituting this value allows the constant of integration to be calculated.
  • This process converts a general solution into a unique solution.
  • Boundary conditions commonly appear in kinematics and modelling questions.

📐 IA Spotlight

  • Boundary conditions are essential when reconstructing models from real-world data.
  • They allow predictions to be anchored to observed values.
  • Discussing assumptions behind boundary conditions strengthens IA evaluation.

📌 3. Definite integrals and area

  • A definite integral calculates accumulated change between two bounds.
  • The Fundamental Theorem of Calculus allows evaluation using antiderivatives.
  • If the function lies above the x-axis, the result represents area.
  • If the function crosses the x-axis, negative contributions must be considered.
  • IB questions often require splitting integrals at intercepts.

🌍 Real-World Connection

  • Velocity–time graphs use integration to calculate displacement.
  • Economics uses integration to measure total cost or revenue from marginal functions.
  • Physics uses definite integrals to calculate work done by variable forces.

🔍 TOK Perspective

  • Integration assumes continuity in real-world phenomena.
  • This raises questions about modelling discrete data with continuous mathematics.
  • To what extent do mathematical assumptions shape knowledge claims?

📌 Exam-Style Questions

Multiple Choice Questions

MCQ 1.
Which statement best explains the purpose of the constant of integration?A. It corrects numerical errors

B. It represents lost information during differentiation

C. It ensures continuity

D. It adjusts limits

Answer: B

Explanation: Differentiation removes constants, so integration must reintroduce them.

MCQ 2.
What does a definite integral represent?A. Gradient

B. Area or accumulation

C. Instantaneous change

D. Curvature

Answer: B

Short Answer Questions

SAQ 1. Why must +C be included in indefinite integrals?

Because infinitely many functions share the same derivative, and the constant accounts for this family.

SAQ 2. Why is integration considered an accumulation process?

It combines infinitely small contributions over an interval to produce a total quantity.

Long Answer / Explainer Questions

LAQ 1.
A particle moves with velocity v(t)=3t²−6t.
(a) Find the displacement function.
(b) Given s(1)=4, find the position function.
(c) Interpret the result.

(a) Integrating gives s(t)=t³−3t²+C.
(b) Substituting s(1)=4 gives C=6.
(c) The constant represents the initial position of the particle.

LAQ 2.
Explain how definite integrals are used to calculate total distance travelled when velocity changes direction.

The velocity function must be analysed for sign changes.
The interval is split at points where velocity equals zero.
The absolute values of the integrals are summed to obtain total distance.