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© Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules 26: Integration by Substitution Part 1 Part 1.

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Presentation on theme: "© Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules 26: Integration by Substitution Part 1 Part 1."— Presentation transcript:

1 © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules 26: Integration by Substitution Part 1 Part 1

2 Integration by Substitution Part 1 "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages" Module C3 AQA Edexcel OCR Module C4 MEI/OCR

3 Integration by Substitution Part 1 Integration by substitution can be used for a variety of integrals: some compound functions, some products and some quotients. Sometimes we have a choice of method.

4 Integration by Substitution Part 1 e.g. 1 Solution: We have already met this type of integral. See if you can do it. Reversing the chain rule gives We’ll use this example to illustrate integration by substitution but if you got it right you can continue to use the reverse chain rule ( also called inspection ).

5 Integration by Substitution Part 1 e.g. 1 Let Method: We must substitute for x and dx. Differentiate: Find dx by treating like a fraction Define u as the inner function Substitute for the inner function...

6 Integration by Substitution Part 1 Let e.g. 1 Differentiate: Method: We must substitute for x and dx. Substitute for the inner function... Define u as the inner function and dx

7 Integration by Substitution Part 1 Let e.g. 1 Differentiate: Method: We must substitute for x and dx. Substitute for the inner function... Define u as the inner function and dx

8 Integration by Substitution Part 1 Let e.g. 1 Differentiate: Method: We must substitute for x and dx. Substitute for the inner function... Define u as the inner function and dx Integrate: Replace u :

9 Integration by Substitution Part 1 Let e.g. 1 Differentiate: Method: We must substitute for x and dx. Substitute for the inner function... Define u as the inner function and dx Integrate: Replace u :

10 Integration by Substitution Part 1 Let e.g. 1 Differentiate: Method: We must substitute for x and dx. Substitute for the inner function... Define u as the inner function and dx Integrate: Replace u :

11 Integration by Substitution Part 1 Students studying the OCR specification should skip the next slide. SKIP

12 Integration by Substitution Part 1 e.g. 2 This is a product but we can’t use integration by parts. Why not? ANS: is a compound function with inner function non-linear. We can’t integrate it. We substitute as before, but using the inner function of the 2 nd factor in the product. If we chose to integrate x instead, at the next stage we would have a more complicated integral than the one we started with.

13 Integration by Substitution Part 1 Let e.g. 2 Differentiate: Substitute for the inner function and dx Define u as the inner function: Cancel the extra x If x won’t cancel we will have to make an extra substitution. We’ll do an example later.

14 Integration by Substitution Part 1 where Integrate: Substitute back: So,

15 Integration by Substitution Part 1 SUMMARY e.gs. Differentiate the substitution expression and rearrange to find dx Method: Substitute for the inner function and dx Define u as the inner function If there’s an extra x, cancel it If x won’t cancel, rearrange the substitution expression to find x and substitute for it Substitution can be used for a variety of integrals Integrate Substitute back

16 Integration by Substitution Part 1 Exercises Use substitution to integrate the following. (Where possible, you could also use a 2 nd method.)

17 Integration by Substitution Part 1 Solutions: 1. Let So,

18 Integration by Substitution Part 1 Solutions: 2. Let So,

19 Integration by Substitution Part 1 3.Solutions: Let So,

20 Integration by Substitution Part 1 Definite integration We work in exactly the same way BUT we must also substitute for the limits, since they are values of x and we are changing the variable to u. A definite integral gives a value so we never return to x.

21 Integration by Substitution Part 1 e.g. 1 Let Limits: So,

22 Integration by Substitution Part 1 So,

23 Integration by Substitution Part 1 e.g. 2 Let Limits: So,

24 Integration by Substitution Part 1 where So, You will often see this written as We leave answers in the exact form.

25 Integration by Substitution Part 1 Exercises Give exact answers.

26 Integration by Substitution Part 1 Let Limits: 1. Solutions: So

27 Integration by Substitution Part 1 Let 2. Limits: We can use the log laws to simplify this. So,

28 Integration by Substitution Part 1 In the next examples, the extra x doesn’t conveniently cancel so we need to substitute for it.

29 Integration by Substitution Part 1 Another product. This time we could use integration by parts but instead we’ll use substitution. Let e.g. 3 Differentiate: Substitute for the inner function and dx Define u as the inner function: The extra x doesn’t cancel so we must substitute for it. Using So,

30 Integration by Substitution Part 1 Can you spot the important difference between these? Ans: We can easily multiply out the brackets in the 2 nd ( where ) Integrate: Replace u : So,

31 Integration by Substitution Part 1 Tip: Don’t be tempted to substitute for the extra x... until you’ve checked to see if it cancels. e.g. 4 Let

32 Integration by Substitution Part 1 e.g. 4 Let So, x doesn’t cancel so now substitute: A multiplying constant... can be taken outside the integral.

33 Integration by Substitution Part 1 So, where

34 Integration by Substitution Part 1 So, where

35 Integration by Substitution Part 1 So, where Remove the brackets and substitute for u :

36 Integration by Substitution Part 1 Exercise Use substitution to integrate the following:

37 Integration by Substitution Part 1 Let So, Solution:

38 Integration by Substitution Part 1

39 The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

40 Integration by Substitution Part 1 SUMMARY e.gs. Differentiate the substitution expression and rearrange to find dx Method: Substitute for the inner function and dx Define u as the inner function If there’s an extra x, cancel it If x won’t cancel, rearrange the substitution expression to find x and substitute for it Substitution can be used for a variety of integrals Integrate Substitute back

41 Integration by Substitution Part 1 Let e.g. 1 Differentiate: Method: We must substitute for x and dx. Substitute for the inner function Define u as the inner function and dx Integrate: Replace u :

42 Integration by Substitution Part 1 Let e.g. 2 Differentiate: Substitute for the inner function and dx Define u as the inner function: Cancel the extra x Sometimes x won’t cancel and we have to make an extra substitution

43 Integration by Substitution Part 1 where Integrate: Replace u : So,

44 Integration by Substitution Part 1 Let e.g. 3 Differentiate: Substitute for the inner function and dx Define u as the inner function: The extra x doesn’t cancel so we must substitute for it. Using So,

45 Integration by Substitution Part 1 Now we can easily multiply out the brackets ( where ) Integrate: Replace u : So,

46 Integration by Substitution Part 1 Definite integration We work in exactly the same way BUT we must also substitute for the limits, since they are values of x and we are changing the variable to u. A definite integral gives a value so we never return to x.

47 Integration by Substitution Part 1 e.g. 1 Let Limits: So,

48 Integration by Substitution Part 1 where So, You will often see this written as


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