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Higher Maths · Worksheet 8

Integration Worksheet | Higher Maths

Skills covered: Indefinite integrals; fractional and negative powers; standard composite-function integrals; trigonometric integrals; definite integrals; initial conditions.

18 questions61 marks3 graded sectionsFree PDF

Download PDF Integration revision notes

Instructions: Attempt all questions. Show sufficient working to justify your answers. Give exact values unless a degree of accuracy is stated. Each section is separate, so you can target the part you need.

These are the same worksheets our pupils use at the Pollokshields tuition centre. Start with Section A if integration still feels shaky; if you are confident, go straight to Section C and time yourself.

Section A

Prerequisite core skills

Revise earlier learning

Question 1

[3 marks]
Find ∫(6x2−4x+3) dx\displaystyle\int (6x^2 - 4x + 3)\,dx.

Question 2

[3 marks]
Find ∫5x dx\displaystyle\int 5\sqrt{x}\,dx, where x>0x > 0.

Question 3

[3 marks]
Find ∫(4x3+x2)dx\displaystyle\int \left(\frac{4}{x^3} + x^2\right) dx, where x≠0x \neq 0.

Question 4

[2 marks]
Find ∫3sin⁡x dx\displaystyle\int 3\sin x\,dx, where xx is in radians.

Question 5

[2 marks]
Find ∫4cos⁡x dx\displaystyle\int 4\cos x\,dx, where xx is in radians.

Question 6

[3 marks]
Find ∫(2x−3)4 dx\displaystyle\int (2x - 3)^4\,dx.
Section B

Topic practice

Build individual techniques

Question 7

[3 marks]
Evaluate ∫02(3x2+2x) dx\displaystyle\int_0^2 (3x^2 + 2x)\,dx.

Question 8

[3 marks]
Evaluate ∫13(6x−4) dx\displaystyle\int_1^3 (6x - 4)\,dx.

Question 9

[3 marks]
Find ∫cos⁡3x dx\displaystyle\int \cos 3x\,dx.

Question 10

[3 marks]
Find ∫sin⁡(2x+π3)dx\displaystyle\int \sin\left(2x + \frac{\pi}{3}\right) dx.

Question 11

[4 marks]
Given f′(x)=6x2−4xf'(x) = 6x^2 - 4x and f(1)=5f(1) = 5, find an expression for f(x)f(x).

Question 12

[4 marks]
A curve satisfies dydx=3x2−12x+5\dfrac{dy}{dx} = 3x^2 - 12x + 5 and passes through (2,7)(2, 7). Find its equation.

Question 13

[4 marks]
Evaluate ∫01(2x+1)3 dx\displaystyle\int_0^1 (2x + 1)^3\,dx.

Question 14

[3 marks]
Given ∫0a2x dx=25\displaystyle\int_0^a 2x\,dx = 25, where a>0a > 0, find aa.
Section C

Exam-style and challenge

Combine skills, exam-style

Question 15

[4 marks]
Given ∫1a3x2 dx=7\displaystyle\int_1^a 3x^2\,dx = 7, where a>1a > 1, find aa.

Question 16

[4 marks]
Find ∫(x−2x)dx\displaystyle\int \left(\sqrt{x} - \frac{2}{\sqrt{x}}\right) dx, where x>0x > 0.

Question 17

[6 marks]
Given f′′(x)=12x−6f''(x) = 12x - 6, f′(0)=4f'(0) = 4 and f(0)=−1f(0) = -1, find f(x)f(x).

Question 18

[4 marks]
Evaluate exactly ∫0π2(2sin⁡x+cos⁡x) dx\displaystyle\int_0^{\frac{\pi}{2}} (2\sin x + \cos x)\,dx.

Looking for the answers?

We do not publish the worked answers online. If you get stuck, bring the worksheet to a session and one of our Higher Maths tutors will go through the method with you, line by line, the way the SQA marking instructions award it.

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Frequently asked questions

Are the answers to these worksheets available online?
No. We keep the worked answers off the website on purpose. Pupils learn far more by attempting every question, checking against their notes and then going over the method with a tutor. Our students work through the full worked solutions in class, and if you are stuck on a particular question you can book a session and we will go through it with you step by step.
Are these official SQA or Qualifications Scotland papers?
No. Every question is newly written by Excel in MathSci to match the Higher Maths course specification. They are original practice resources, not official Qualifications Scotland papers, so use them alongside the real past papers rather than instead of them.
How should I use the three sections?
Section A checks the prerequisite skills you need before the topic makes sense, so start there if the topic feels shaky. Section B builds each technique on its own. Section C is exam-style and combines several skills, so treat it as a timed test once Sections A and B feel comfortable.
Can I print the worksheet?
Yes. Download the free PDF, which has space to write your working under every question, or print this page directly. The worksheet is free for personal study; please ask before reproducing it for a class or a commercial service.

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