Topic 5.18 · HL only

Partial fractions in integration: notes and practice questions

Q: Is Partial fractions in integration SL or HL?

Partial fractions in integration is HL only. SL students are not examined on it.

Q: How do I revise Partial fractions in integration for IB Maths AA?

Start from the core idea: partial fractions are used to integrate rational functions where direct substitution fails. In the exam: recognising which standard form an integral matches is the skill, and completing the square to get there (x^2 + 2x + 5 → (x+1)^2 + 4) is the standard step. Paper 1. Then practise exam-style questions, easiest first, writing out every step of your working before you check it.

Q: How does FourtyFive help me practise Partial fractions in integration?

FourtyFive has 7 Partial fractions in integration questions. Every answer you write is marked mark by mark, IB-style, and you see where each mark was won or lost. Every part has a hint, the AI tutor helps you through the step you are stuck on, and your Study Profile picks what to practise next.

Q: Is FourtyFive free for Partial fractions in integration practice?

Yes. A free account gives you 50 marked answers a month, and you do not need a card to sign up.

Q: Can I handwrite Partial fractions in integration answers on an iPad?

Yes. In the FourtyFive iPad app you write your working by hand with Apple Pencil, the way you would on paper, and it is marked the same way.

2 min read Jump to 7 practice questions

Summary

Partial fractions are used to integrate rational functions where direct substitution fails.
First, ensure the fraction is proper (degree of numerator < degree of denominator) using polynomial long division if necessary.
Factorize the denominator into linear factors (distinct or repeated).
Decompose the rational function into a sum of simpler fractions based on the factors.
Determine the unknown constants (A, B, C...) using substitution (cover-up method) or by equating coefficients.
Integrate each resulting term using logarithmic or power rules, and potentially the arctan rule for irreducible quadratics.

How it is examined

Recognising which standard form an integral matches is the skill, and completing the square to get there ( $x^2 + 2x + 5 \to (x+1)^2 + 4$ ) is the standard step. Paper 1. 4 to 7 marks.

Given in the booklet

The full HL derivative and integral tables are given, including the $\arctan$ and $\arcsin$ integral forms $\int \frac{1}{a^2+x^2}\mathrm{d}x$ and $\int \frac{1}{\sqrt{a^2-x^2}}\mathrm{d}x$ .

Key ideas

Derivatives of $\tan x$ , $\sec x$ , $\csc x$ , $\cot x$ , $a^x$ , $\log_a x$ , $\arcsin x$ , $\arccos x$ , $\arctan x$ .
Indefinite integrals of the derivatives of any of the above functions.
The composites of any of these with a linear function.
Use of partial fractions to rearrange the integrand.

Linking questions

The partial fractions limit from AHL 1.11 applies here too: two distinct linear factors in the denominator, nothing else.

Practice questions

7 questions · 4 medium · 3 hard

Showing 7 of 7

Question 1

MediumPaper 1 · no calculator8 marks

Let $g(x) = \frac{x+8}{x^2+x-2}$ for $x \in \mathbb{R}, x \neq 1, x \neq -2$ .

Find $\int g(x) \, dx$ .

Question 2

HardPaper 1 · no calculator10 marks

Given that $\int_2^3 \frac{9x+1}{3x^2 - x - 2} dx = \ln k$ , find the value of $k$ .

Question 3

MediumPaper 1 · no calculator8 marks

Let $f(x) = \frac{x+8}{x^2+x-6}$ for $x > 2$ .

Use partial fractions to find $\int f(x) \text{ } dx$ .

Question 4

HardPaper 1 · no calculator38 marks

(a)

Find the general solution to the following differential equation. (a)

$\frac{dy}{dx} = \frac{y^2}{x^2-1}$

[5]

(b)

(b)

$\frac{dy}{dx} = y^2 \sin x \cos x$

[5]

(c)

(c) Find the particular solution to the differential equation $(x^2+4) \frac{dy}{dx} = xy$ , given the initial condition $y(0)=1$ .

[6]

(d)

(d)

$e^x \frac{dy}{dx} = \frac{1}{y}$

[4]

(e)

(e)

$\frac{dy}{dx} + 2y = xe^{-x}$

[6]

(f)

(f) $x \frac{dy}{dx} - 3y = x^5$ for $x>0$ .

[6]

(g)

(g) $\frac{dy}{dx} + y \cot x = \cos x$ for $0 < x < \pi$ .

[6]

Question 5

MediumPaper 1 · no calculator7 marks

(a)

Express $x^2 + 4x + 8$ in the form $(x-p)^2 + q$ , where $p, q \in \mathbb{Z}$ .

[2]

(b)

Hence, find the exact value of $∫_{-2}^0 \frac{3}{x^2+4x+8} \text{d}x$ .

[5]

Question 6

HardPaper 1 · no calculator15 marks

(a)

Consider the function $f(x) = \frac{4x-2}{x^2-x-2}$ , for $x \in \mathbb{R}, x \neq 2, x \neq -1$ .

(a) Express $x^2-x-2$ in the form $(x+h)^2+k$ .

[2]

(b)

(b) Express $f(x)$ in partial fractions.

[3]

(c)

[5]

(d)

(d) Find the area of the region enclosed by the graph of $y = f(|x|)$ , the x-axis and the lines with equations $x = -4$ and $x = -3$ .

[5]

Question 7

MediumPaper 1 · no calculator7 marks

By using the substitution $u = e^x$ , find $\int \frac{e^x}{e^{2x} - 2e^x - 8} \mathrm{d}x$ .

Every Partial fractions in integration question, marked for you

Every answer is marked mark by mark, IB-style, and the AI tutor helps when you are stuck.

Where marks are lost

Rounding an intermediate value and then using it.
Answering to the wrong accuracy. Two significant figures, or six, where the rule says exactly or three.
Writing the answer and nothing else.

FAQ

Questions,
answered.

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What does Partial fractions in integration cover in IB Maths AA?

Partial fractions are used to integrate rational functions where direct substitution fails. First, ensure the fraction is proper (degree of numerator < degree of denominator) using polynomial long division if necessary. Factorize the denominator into linear factors (distinct or repeated).

Is Partial fractions in integration SL or HL?

Partial fractions in integration is HL only. SL students are not examined on it.

How do I revise Partial fractions in integration for IB Maths AA?

Start from the core idea: partial fractions are used to integrate rational functions where direct substitution fails. In the exam: recognising which standard form an integral matches is the skill, and completing the square to get there (x^2 + 2x + 5 → (x+1)^2 + 4) is the standard step. Paper 1. Then practise exam-style questions, easiest first, writing out every step of your working before you check it.

How does FourtyFive help me practise Partial fractions in integration?

FourtyFive has 7 Partial fractions in integration questions. Every answer you write is marked mark by mark, IB-style, and you see where each mark was won or lost. Every part has a hint, the AI tutor helps you through the step you are stuck on, and your Study Profile picks what to practise next.

Is FourtyFive free for Partial fractions in integration practice?

Yes. A free account gives you 50 marked answers a month, and you do not need a card to sign up.

Can I handwrite Partial fractions in integration answers on an iPad?

Yes. In the FourtyFive iPad app you write your working by hand with Apple Pencil, the way you would on paper, and it is marked the same way.

Partial fractions in integration: notes and practice questions

How it is examined

Linking questions

Practice questions

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Every Partial fractions in integration question, marked for you

Where marks are lost

Practise this topic as a session

Questions,
answered.

Start with the IB question
bank built for you.

How it is examined

Linking questions

Practice questions

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Every Partial fractions in integration question, marked for you

Where marks are lost

Questions,answered.

Start with the IB questionbank built for you.

Questions,
answered.

Start with the IB question
bank built for you.