Radford Mathematics IB Mathematics resources • AA HL

IB Mathematics: Analysis and Approaches HL — Topic 5 Calculus

Anti-Differentiation and Boundary Conditions

Reverse the power rule, include the constant of integration and use a point to determine a particular function.

AA HL · SL 5.5

Learning goal

Find indefinite integrals of polynomial functions and determine the constant using a boundary condition.

Syllabus link

AA HL: SL 5.5 · Topic 5 Calculus.

Big idea

An indefinite integral is a family of functions; the constant of integration represents the vertical separation between members of that family.

Key relationship

\(\int ax^n\,dx=\frac{a}{n+1}x^{n+1}+C\;(n\ne-1)\)

1

Core idea

An indefinite integral is a family of functions; the constant of integration represents the vertical separation between members of that family.

Find indefinite integrals of polynomial functions and determine the constant using a boundary condition.

2

Key results and method

\[\int ax^n\,dx=\frac{a}{n+1}x^{n+1}+C\;(n\ne-1)\]
\[F\'(x)=f(x)\iff F(x)=\int f(x)\,dx\]
  1. Increase each power by one.
  2. Divide by the new power.
  3. Add \(+C\).
  4. Substitute the boundary condition and solve for \(C\).
3

Worked example

Question

If \(f\'(x)=6x^2-4x\) and \(f(1)=5\), find \(f(x)\).

Solution
  1. \(f(x)=2x^3-2x^2+C\).
  2. \(5=2-2+C\Rightarrow C=5\).
Answer: \(f(x)=2x^3-2x^2+5\).
4

Practice

Try these questions before moving on.

  1. Integrate \(3x^4-2x^{-2}\).
  2. Find a curve given its gradient function and one point.
  3. Differentiate your final answer to check it.

Search the worksheet library for more practice →

5

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