Radford Mathematics IB Mathematics resources • AA HL

IB Mathematics: Analysis and Approaches HL — Topic 5 Calculus

Further Derivatives and Antiderivatives

Extend the standard derivative/integral toolkit and use partial fractions to create integrable forms.

AA HL · AHL 5.15

Learning goal

Differentiate and integrate the additional AHL functions and rearrange integrands with partial fractions.

Syllabus link

AA HL: AHL 5.15 · Topic 5 Calculus.

Big idea

The HL toolkit expands the library of recognised forms; algebraic rearrangement is often the gateway to a standard integral.

Key relationship

\((\tan x)\'=\sec^2x\)

1

Core idea

The HL toolkit expands the library of recognised forms; algebraic rearrangement is often the gateway to a standard integral.

Differentiate and integrate the additional AHL functions and rearrange integrands with partial fractions.

2

Key results and method

\[(\tan x)\'=\sec^2x\]
\[\int\sec^2x\,dx=\tan x+C\]
\[(a^x)\'=a^x\ln a\]
\[(\arctan x)\'=\frac1{1+x^2}\]
\[\int\frac1{1+x^2}dx=\arctan x+C\]
  1. Identify a standard derivative pair.
  2. For rational functions, check whether partial fractions are appropriate.
  3. Complete the square when inverse-trigonometric forms are present.
  4. Differentiate to check the final antiderivative.
3

Worked example

Question

Evaluate \(\int\frac{1}{x^2+2x+5}dx\).

Solution
  1. Complete the square: \(x^2+2x+5=(x+1)^2+4\).
  2. Use the arctangent standard form.
Answer: \(\frac12\arctan\frac{x+1}{2}+C\).
4

Practice

Try these questions before moving on.

  1. Integrate a secant-squared linear composite.
  2. Differentiate an inverse trigonometric function.
  3. Use partial fractions to integrate a rational function with two distinct linear factors.

Search the worksheet library for more practice →

5

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