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Unsure about Integration by Substitution?

Unsure about Integration by Substitution?

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Covers the whole of the AH Maths course

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Resources used with students in Scottish Secondary Schools

Integration by Substitution

Welcome to advancedhighermaths.co.uk

A sound understanding of Integration by Substitution is essential to ensure exam success.

Study at Advanced Higher Maths level will provide excellent preparation for your studies when at university. Some universities may require you to gain a pass at AH Maths to be accepted onto the course of your choice. The AH Maths course is fast paced so please do your very best to keep on top of your studies.

For students looking for extra help with the AH Maths course in 2019/20 you may wish to consider subscribing to the fantastic additional exam focused resources available in the Online Study Pack.

We hope you find this website useful and wish you the very best with your studies in 2019/20.

Please find below:

1. Links and Exam Formulae

2. Integration Resources

3. Further Integration Resources

4. MIA Text Book Questions – Integration

5. MIA Text Book Questions – Further Integration

6. About Integration by Substitution

7. AH Maths Past and Practice Papers

8. AH Maths Past Paper Questions by Topic

9. Recommended AH Book

10. Exam Focused Online Study Pack – For students looking for a ‘good’ Pass

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Advanced Higher Maths Resources

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1. Links and Exam Formulae

AH Maths - Home Page LinkHERE
AH Maths - Whole Course LinkHERE
AH Maths - Exam Formula ListHERE
Higher Maths - Exam Formulae ListHERE
AH Maths - Complete Formula ListHERE
SQA Full Course Content and OutlineHERE

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2. Integration Resources

Thanks to the author(s) for making these available.

Integration Theory Guide HERE
Integration HSN Summary NotesHERE

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3. Further Integration Resources

Thanks to the author(s) for making these available.

Further Integration Theory Guide 1HERE
Further Integration Theory Guide 2HERE
Further Integration HSN Summary NotesHERE

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4. MIA Text Book Questions – Integration

Recommended questions from the new MIA Text Book are shown below.

Worked solutions to all questions below are available in the Online Study Pack.

Subtopic
______________________________________		Page No
__________		Exercise
___________		Recommended Questions
_____________________
Integration (Higher Revision)Page 100Exercise 7.1Q1a-i,2a-i,3a-l,4a-f
Integration by SubstitutionPage 103Exercise 7.2Q1a,c,e,g,i,k,m,o,q,s,u,w
Integration by Substitution - Extra Revision!Page 103Exercise 7.2Q1b,d,f,h,j,l,n,p,r,t,v,x
Further Integration by SubstitutionPage 105Exercise 7.3Q2a,b,c,d,4a,b,c,d
Further Integration by SubstitutionPage 105Exercise 7.3Q6a,b,c,d
Further Int'n by Sub'n - sin^m(x), cos^n(x)Page 105Exercise 7.3Q7a,b,c,d,e,f
Further Integration by Substitution - logsPage 105Exercise 7.3Q11a,b,c,d
Substitution & Definite IntegralsPage 107Exercise 7.4Q1a,c,e,g,i,k
Area between curve & x-axisPage 120Exercise 7.10Q1,3
Area between curve & y-axisPage 120Exercise 7.10Q6,7
Volume - revolved around x-axis SQA Question2014 Q10 (SQA)
Volume - revolved around y-axis SQA Question2017 Q16 (SQA)
Volume - revolved around x-axisPage 120Exercise 7.10Q11,12
Applications of Integral CalculusPage 187Exercise 11.1Q4,14

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5. MIA Text Book Questions – Further Integration

Recommended questions from the new MIA Text Book are shown below.

Worked solutions to all questions below are available in the Online Study Pack.

Subtopic
______________________________________		Page No
__________		Exercise
_________________		Recommended Questions
__________________________
Integration using Inverse Trig FunctionsPage 111Exercise 7.6Q1,2,3,4a,b
Integration using Partial FractionsPage 113Exercise 7.7Q1a,b,2a,b,3a,b,4a,b,5a,b,6a,b
Integration by Parts - 1Page 116Exercise 7.8Q1a-l
Integration by Parts - 2Page 116Exercise 7.8Q2a,c,d,e,f,g,h
Integration by Parts - 3Page 116Exercise 7.8Q5a,b,Q6a,b
Integration by Parts - Special Cases - 1Page 118Exercise 7.9 Q1a,b,c,d
Integration by Parts - Special Cases - 2Page 118Exercise 7.9Q2a,b,c,d,e
First Order Diff Eqns - General SolnPage 128Exercise 8.1Q1a-j
First Order Diff Eqns - Particular SolnPage 128Exercise 8.1Q2a-g
Differential Equations in ContextPage 131Exercise 8.2Q2,4,5,6

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6. About Integration by Substitution

For a more detailed explanation, please first read the Theory Guides above.

Integration by substitution

  • is useful when the derivative of one part of the integrand is related to another part of the integrand
  • involves rewriting the entire integral (including the ” dx ” and any limits) in terms of another variable before integrating

Note that the answer for an indefinite integral must always be expressed in terms of the original variable.

Example

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Exam Question

Source: SQA AH Maths Paper 2017 Question 6

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7. AH Maths Past & Practice Papers

Thanks to the SQA for making these available.

Worked solutions to SQA Exam Questions available in the Online Study Pack.

Year
____		Paper Type
_________________		Exam Paper
______________		Marking Scheme
_______________________________________
2019AH SpecimenSpecimenMarking Scheme
2019Advanced HigherExam PaperMarking Scheme
2018Advanced HigherExam PaperMarking Scheme
2017Advanced HigherExam PaperMarking Scheme
2016Advanced HigherExam PaperMarking Scheme
2016AH SpecimenSpecimenMarking Scheme
2016AH ExemplarExemplarMarking Scheme
2015Advanced HigherExam PaperMarking Scheme
2014Advanced HigherExam PaperMarking Scheme
2013Advanced HigherExam PaperMarking Scheme
2012Advanced HigherExam PaperMarking Scheme
2011Advanced HigherExam PaperMarking Scheme
2010Advanced HigherExam PaperMarking Scheme
2009Advanced HigherExam PaperMarking Scheme
2008Advanced HigherExam PaperMarking Scheme
2007Advanced HigherExam PaperMarking Scheme
2006Advanced HigherExam PaperMarking Scheme
2005Advanced HigherExam PaperMarking Scheme
2004Advanced HigherExam PaperMarking Scheme
2003Advanced HigherExam PaperMarking Scheme
2002Advanced HigherExam PaperMarking Scheme
2001Advanced HigherExam Paper

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8. AH Maths Past Paper Questions by Topic

Thanks to the SQA for making these available.

Questions and answers have been split up by topic for ease of reference.

Worked solutions to all SQA questions below available in the Online Study Pack.

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Paper
___________		.
Marking
______		Binomial
Theorem
________		Partial
Fractions
________		.
Differentiation
___________		Further Differentiation
___________		.
Integration
___________		Further
Integration
____________		Functions
& Graphs
___________		Systems of
Equations
____________		Complex
Numbers
__________		Seq &
Series
_________		Further Seq
& Series
____________		.
Matrices
_________		.
Vectors
__________		Methods
of Proof
__________		Further No
Theory
___________		Differential
Equations
____________		Further
Differential Eqns
_________________
Specimen P1MarkingQ2Q4Q6Q8Q3Q5Q1Q7
Specimen P2MarkingQ3Q1Q2,4,8,10Q7Q11Q5Q13Q9Q6Q12
2019MarkingQ9Q4Q1a,b,6Q1c,5,10Q16bQ16aQ3Q18Q7,17Q2Q15Q11,14Q12Q13Q8
2018MarkingQ3Q2Q1bQ1a,c,6,13Q8Q15aQ16aQ4,10Q14Q17Q7,11Q16Q9,12Q5Q15b
2017MarkingQ1Q2Q3Q11,18Q16Q6Q12Q5Q17Q4,10Q7Q15Q13Q8Q9Q14
2016MarkingQ3Q13Q1a,bQ1c,11Q13Q9Q12Q4Q8Q2Q6Q7Q14Q5,10Q16Q15
2015MarkingQ1,9Q2Q4,6,8Q17Q10Q14Q13Q3Q5,11Q15Q12Q7Q18Q16
2014MarkingQ214bQ1,13Q1,4,6Q10,12Q15Q11Q3Q16Q14Q9Q7Q5Q7Q8
2013MarkingQ1Q2Q11Q4,6Q8Q13Q7,10Q17Q3Q15Q9,12Q5Q16Q14
2012MarkingQ415aQ1Q12,13Q8Q11Q7Q14Q3,16bQ2Q6Q9Q516aQ10Q15
2011MarkingQ2Q13b,73aQ1,11aQ1,11,16Q6Q10Q8,13Q5Q4Q15Q12Q9Q14
2010MarkingQ5Q1Q13Q15Q3,7Q10Q16Q2Q9Q4,14Q6Q8,12Q11
2009MarkingQ8Q14Q1aQ1b,11Q5,7Q9Q1316aQ6Q12Q14Q2Q16Q4Q10Q3Q15
2008MarkingQ8Q4Q10,15Q2,5Q4,9,10Q7Q3Q16Q1Q12Q6Q14Q11Q13
2007MarkingQ1Q4Q2Q13Q4,10Q4Q16Q3,11Q9Q6Q5Q15Q12Q7Q14Q8
MixedMixedMixedMixedMixedMixedMixedMixedMixedMixedMixedMixedMixedMixedMixedMixedMixed

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9. AH Maths Text Book – Maths In Action (2nd Edition) by Edward Mullan 

A fully revised course for the new Curriculum for Excellence examination that is designed to fully support the course’s new structure and unit assessment. A part of the highly regarded Maths in Action series, it provides students with a familiar, clear and carefully structured learning experience that encourages them to build confidence and understanding.
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10. Exam focused Online Study Pack 

Through step-by-step worked solutions to exam questions available in the Online Study Pack we cover everything you need to know about Integration by Substitution to pass your final exam.

As well as students studying Advanced Higher Mathematics, the resources will benefit young adults studying A-Level Mathematics and undergraduates who need a little extra help. Particular benefit will be to students who have gained a ‘Conditional’ University place and are therefore required to pass in order to gain entry onto the course of their choice.

Subscribing to the Online Study Pack may therefore be one of your best ever investments. Please click Online Study Pack to view screenshots, examples and instructions on how to subscribe.

Please speak with your mum/dad and show them our fantastic website.

We hope the resources on this website prove useful and wish you the very best of success with your AH Maths course in 2019/20.

Get the Study Pack - just £9.99

Teachers click here >

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