Topic 3 - Integation
We have below bronze, silver, gold and platinum tutorials for our Advanced Higher Maths Integration topic. Typically as you work through the modules, the difficulty of the skills increase by one step ensuring smooth progression. Simply click on the video to play it and pause it at the relevant points for you to attempt your practice question. Each tutorial has a corresponding worksheet for you to download for additional practice. Make sure you have a pen and paper handy! Be sure to mark your work for every exercise attempted. Good luck!
Essential prior knowledge:-
Higher Maths
- 9.1 - Indefinite integrals at bronze, silver and gold level.
- 9.2 - Definite integrals at bronze, silver and gold level.
- 13.2 - Further integration at bronze, silver and gold level.
Contents:-
3.0 - An introduction to integration
3.1 - Integrating inverse trig functions
3.2 - Integrating rational functions
3.3 - Integration by substitution
3.4 - Integration by parts
3.1: Integrating inverse trig functions
Outcome 1 - Bronze Level
Video 36: Integrating inverse trig functions. In the previous topic, we learned how to differentiate inverse trig functions. In this video we will learn how to undo this process.
Now you've watched the video, try these practice questions.
Adobe Acrobat document [103.1 KB]
Outcome 2 - Silver Level
Video 37: Integrating inverse trig functions with constants. In this video we learn how to isolate constants to help ensure our integration is as straight forward as possible.
Now you've watched the video, try these practice questions.
Adobe Acrobat document [113.5 KB]
Outcome 3 - Gold Level
Video 38: Integrating inverse trig functions with substitution. We have learned that these formulae only work for x2. Now we will learn a clever substitution technique to help us deal with more than one x2.
Now you've watched the video, try these practice questions.
Adobe Acrobat document [123.8 KB]
3.2: Integrating rational functions
Outcome 1 - Bronze Level
Video 39: Integrating rational functions. In this video we will learn about a new standard integral for rational functions and how we can use this to help us integrate related functions.
Now you've watched the video, try these practice questions.
Adobe Acrobat document [103.3 KB]
Outcome 2 - Silver Level
Video 40: Integrating rational functions when the denominator cannot be facorised. In this video we will learn how to split up rational functions as separate fractions to allow us to integrate.
Now you've watched the video, try these practice questions.
Adobe Acrobat document [119.5 KB]
Outcome 3 - Gold Level
Video 41: Integrating rational functions when the denominator can be facorised. In this video we will have to recall our knowledge of partial fractions to enable us to integrate rational functions where the denominator can be factorised.
Now you've watched the video, try these practice questions.
Adobe Acrobat document [116.1 KB]
SQA Exam Questions
You will find below a collection of Integrating Rational Functions SQA past paper questions. You should click on a question, pause the video when instructed and give the question your best shot. Then un-pause the video and watch the worked solution. You should make this section a regular place to come back to when revising. Even if you get the correct answer, it is recommended that you try all these questions more than once at various stages throughout the year.
3.2 - Integrating Rational Functions
- 2018 Question 2
- 2016 Question 13
- 2015 Question 17
- 2013 Question 6
- 2011 Question 1
- 2010 Question 7
- 2009 Question 5
- 2008 Question 4
- 2007 Question 4
Alternatively you can click here to view all the questions on a playlist.
3.3: Integration by Substitution
Outcome 1 - Bronze Level
Video 42: Integration by substitution for indefinite integrals. In this video we learn how to "undo" the Chain Rule where we learn to integrate expressions that contain functions with functions.
Now you've watched the video, try these practice questions.
Adobe Acrobat document [111.0 KB]
Outcome 2 - Silver Level
Video 43: Integration by substitution for definite integrals. In this video we learn how to evaluate definite integrals by subsitution.
Now you've watched the video, try these practice questions.
Adobe Acrobat document [124.1 KB]
Outcome 3 - Gold Level
Video 44: Integration by substitution with an extra 'x' term. How can we integrate when, after our substitution, we are still left with 2 variables? Find out how in this video!
Now you've watched the video, try these practice questions.
Adobe Acrobat document [127.9 KB]
Outcome 4 - Platinum Level
Video 45: Integration by substitution with trig identities. In this video we learn how to answer a true level 'A' question by using our knowledge of trig identities to enable us to integrate using substitution.
Now you've watched the video, try these practice questions.
Adobe Acrobat document [132.0 KB]
SQA Exam Questions
You will find below a collection of Integration by substitution SQA past paper questions. You should click on a question, pause the video when instructed and give the question your best shot. Then un-pause the video and watch the worked solution. You should make this section a regular place to come back to when revising. Even if you get the correct answer, it is recommended that you try all these questions more than once at various stages throughout the year.
3.3 - Integration by substitution
- 2018 Question 8
- 2017 Question 6
- 2014 Question 12
- 2012 Question 8
- 2011 Question 11
- 2010 Question 3(a)
- 2009 Question 7
- 2008 Question 9
- 2007 Question 10
- 2005 Question 5
- 2004 Question 9
- 2003 Question 5
- 2002 Question 6
Alternatively you can click here to view all the questions on a playlist.
3.4: Integration by parts
Outcome 1 - Bronze Level
Video 46: Integration by parts. In this video we learn how to integrate two functions multiplied together when substitution won't work.
Now you've watched the video, try these practice questions.
Adobe Acrobat document [114.6 KB]
Outcome 2 - Silver Level
Video 47: Integration by parts more than once. In this video we learn how to integrate by parts where our function 'u' needs a little more simplifying.
Now you've watched the video, try these practice questions.
Adobe Acrobat document [125.3 KB]
Outcome 3 - Gold Level
Video 48: Integration by parts using the integral I. In this video we learn how to integrate by parts when we get "stuck in a loop" using a very clever technique.
Now you've watched the video, try these practice questions.
Adobe Acrobat document [123.4 KB]
Congratulations!
You have completed this module - well done! Remember to go back and redo any areas of weakness, that's how we improve. If you are happy with all areas, go back to the main menu and pick another topic to practice!
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