5.10 Indefinite integrals and substitution by inspection questions for IB Maths AA

65 questions on Indefinite integrals and substitution by inspection, written by Augment in the style of IB exam papers. Six of them are below in full, to answer on paper.

IB Mathematics: Analysis and Approaches
Part of 5 Calculus
SL and HL

Questions by paper and level

SL and HL questions are for students at either level. HL only questions cover what only HL students study.

Paper	SL and HL	HL only	All
Paper 1	35	17	52
Paper 2	11	2	13
Every paper	46	19	65

Counted on 10 November 2026

Questions to try

Written by Augment in the style of IB exams, not past papers. Answer them on paper, then check each one against its markscheme with a free account.

Question 1 · AA-00141Paper 1 · 6 marks

The gradient of a curve is given by $\frac{d y}{d x} = 6 \sin (2 x - \frac{π}{3})$ . The curve passes through the point $(\frac{3 π}{4}, 2 \sqrt{3})$ .

Find $y$ in terms of $x$ .

6 marks

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Question 2 · AA-00744Paper 2 · 7 marks

Let $f (x) = 4 x e^{- x^{2}}$ , for $x \geq 0$ .

y = f(x)

(a)
Find $\int f (x) d x$ .
2 marks
(b)
For $k > 0$ , the region bounded by the curve $y = f (x)$ , the $x$ -axis and the vertical line $x = k$ has area $1.9$ .
Find the value of $k$ .
4 marks
(c)
Explain why no region of this form can have an area of $2$ or more.
1 mark

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Question 3 · AA-00156Paper 1 · 7 marks

Let $g (x) = \sin^{2} x \cos^{2} x$ , for $x \in R$ .

(a)
Show that $8 g (x) = 1 - \cos 4 x$ .
3 marks
(b)
Hence, find the exact value of $\int_{0}^{\frac{π}{8}} g (x) d x$ , giving your answer in the form $\frac{π - a}{b}$ , where $a, b \in Z^{+}$ .
4 marks

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Question 4 · AA-00745Paper 2 · 6 marks

A seedling is planted in a garden. Its height, $h$ cm, $t$ weeks after planting is modelled by a function whose rate of change is

\frac{d h}{d t} = 3 e^{- 0.4 t} + 0.2, t \geq 0.

When the seedling is planted, its height is $4$ cm.

(a)
Find an expression for $h$ in terms of $t$ .
4 marks
(b)
Find the time taken for the seedling to reach a height of $12$ cm.
2 marks

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Question 5 · AA-00164Paper 1 · 17 marks

Let $g (x) = (\sin x + \cos x)^{4}$ , for $x \in R$ .

(a)
Show that $(\sin x + \cos x)^{2} = 1 + \sin 2 x$ .
2 marks
(b)
Hence, show that $g (x) = \frac{3}{2} + 2 \sin 2 x - \frac{1}{2} \cos 4 x$ .
4 marks
(c)
Find the exact value of $\int_{0}^{\frac{π}{4}} g (x) d x$ .
5 marks
(d)
Find the range of $g$ .
2 marks
(e)
Solve the equation $g (x) = \frac{9}{4}$ , for $0 \leq x \leq π$ .
4 marks

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Question 6 · AA-00746Paper 2 · 6 marks

A cup of hot chocolate is left to cool in a room. Its temperature, $T^{\circ} C$ , $t$ minutes after it is poured is modelled by a function with

\frac{d T}{d t} = - 8 e^{- 0.2 t}, t \geq 0.

When it is poured, the temperature of the hot chocolate is $60^{\circ} C$ .

(a)
Find an expression for $T$ in terms of $t$ .
3 marks
(b)
Find the time taken for the hot chocolate to cool to $30^{\circ} C$ .
2 marks
(c)
According to the model, write down the temperature of the room.
1 mark

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