C4 Chapter 6: Integration Dr J Frost ([email protected]) www.drfrostmaths.com Last modified: 9th February 2016 Overview You havent probably seen integration since C1! Youll be able to integrate a significantly greater variety of expressions, and be able to solve differential equations (which we encountered earlier in C4). Integration by standard result (Theres certain expressions youre expected to know straight off.) 2 sec =tan + Integration by substitution (We make a substitution to hopefully make the expression easier to integrate) Let Integration by reverse chain rule (We imagine what would have differentiated to get the expression.) 1 4 sin cos = 4 sin +

3 Integration by parts (Allows us to integrate a product, just as the product rule allowed us to differentiate one) Overview Volumes of revolution (Find the volume of a solid formed by spinning a curve around the axis) = 2 Finding an area using parametric equations (Find the volume of a solid formed by spinning a curve around the axis) 2

= 2 Solving Differential Equations (Solving here means to find one variable in terms of another without derivatives present) = SKILL #1: Integrating Standard Functions Theres certain results you should be able to integrate straight off, by just thinking about the opposite of differentiation. ? ? ? ? ? ? ?

? ? The has to do with problems when is negative (as is not defined in the real domain) Remember my memorisation trick of picturing sin above cos from C3, so that going down is differentiating and going up is integrating, and we change the sign if the wrong way round. Its vital you remember this one. Have a good stare at this slide before turning your paper over lets see how many you remember Quickfire Questions (without cheating!) sectan =sec+ sin =cos + 2 =cot + cos =sin+ ? ?

? ? Quickfire Questions (without cheating!) 2 sec =tan + cot =+ 1 =ln + sin =cos + Quickfire Questions (without cheating!) 2 =cot + sin =cos + sectan =sec+ cos =sin + ? ? ?

? Test Your Understanding 3 2 3 2 ? + 2 cos + =2 sin +3 ln 3 cos ? + sin2 = cosec cot = Hint: What reciprocal trig functions does this simplify to? Exercise 6A 1 Integrate the following. ? ? ?

? ? a c e Calculate the following. g i 2 a ? ? c e g i ? ? ? SKILL #2: Integrating by Reverse Chain Rule ?

Therefore: ? ! For any expression where inner function is , integrate as before and . Quickfire: ? ? ? ? ? ? ? Check Your Understanding ? ? ? ? Exercise 6B

Page 86 1 a c 3 a ? ? ? e f ? g 2 a ? ? c ? d

e ? ? c ? ? f ? h ? ? j ? b b

d ? ? h ? 4 a ? c ? SKILL #3: Integrating using Trig Identities Some expressions, such as and cant be integrated directly, but we can use one of our trig identities to replace it with an expression we can easily integrate. Q Find Q Find Do you know a trig identity involving ?

? ? Q Q Find Find From formula booklet: ? ? Check Your Understanding Q Find Hint: What identity do you know involving ? Recall that ?

Q Find Q 2 ( sec+tan ) ? Bro Tip: The quick way is this. is the sum of the two nums. is the difference. Find From formula booklet: ? Exercise 6C Page 88 1 a

? 2 a b ? c ? c e e i ? h i ? ?

f g g ? d ? ? ? ? ? ? ? ? 3 a c ? e ?

g ? SKILL #4: Using Partial Fractions We saw earlier that we can split some expressions into partial fractions. This allows us to differentiate some expressions with more complicated denominators. Find ? Test Your Understanding Find Find ? ? Exercise 6D Page 92 Use partial fractions to integrate the following. 1 a ?

c ? e ? g ? i ? 2 a c ? ? e ? SKILL #5: Consider and Scale

Theres certain more complicated expressions which look like the result of having applied the chain rule. The process is then simply: 1. Consider some expression that will differentiate to something similar to it. 2. Differentiate, and adjust for any scale difference. 3 2 2 ( +5 ) cos sin The first looks like it arose from differentiating the inside the brackets. 2 2 +1 The probably arose from differentiating the . The probably arose from differentiating the .

Consider Consider Thus Thus Consider Thus ? ? ? SKILL #5: Consider and Scale ! Integration by Inspection: Use common sense to consider some expression that would differentiate to the expression given. Then scale appropriately. Common patterns: In words: If the bottom of a fraction differentiates to give the top (forgetting scaling), try ln of the bottom.

2 3 +1 Consider Consider Therefore: Therefore: ? 2 +1 ?

Quickfire In your head! ? ? ? ? ? Not in your head ( ) = + + 2 3 ? ( + 5) Bro Tip: If theres as power around the whole denominator, DONT use :

reexpress the expression as a product. e.g. Test Your Understanding 5 sin ( cos +1 ) Consider 2 3 ( 2 + cot ) Consider Therefore: ? sec 2 2 tan 2 + 1

1 ln tan 2 +1+ 2 ? ? Thus 5 tan sec 2 Note that , so the power of stays the same. Try Then Thus ? Exercise 6E Page 94 2

? 1 ? ? ? ? ? ? ? ? ? ? ? ? ? SKILL #6: Integration by Substitution For some integrations involving a complicated expression, we can make a substitution to turn it into an equivalent integration that is simpler. We wouldnt be able to use

reverse chain rule on the following: Q Use the substitution to find The aim is to completely remove any reference to , and replace it with . Well have to work out and so that we can replace them. STEP 1: Using substitution, work out and (or variant) STEP 2: Substitute these into expression. 1 =2 = 2 5? = 2 ? Bro Tip: Be careful about ensuring you reciprocate when rearranging. Bro Tip: If you have a constant factor, factor it out of the integral.

STEP 3: Integrate simplified expression. ? STEP 4: Write answer in terms of . ? How can we tell what substitution to use? In Edexcel you will usually be given the substitution! However in some other exam boards, and in STEP, you often arent. Theres no hard and fast rule, but its often helpful to replace to replace expressions inside roots, powers or the denominator of a fraction. Sensible substitution: cos 1+sin 6 2 1+

1 1+ =+? But this can be integrated ? = by inspection. ? =+ = ? + Another Example Q Use the substitution to find STEP 1: Using substitution, work out and (or variant)

STEP 2: Substitute these into expression. Notice this time we didnt find or . We could, but then , and it would be slightly awkward simplifying the expression (although is still very much a valid ? method!) ? STEP 3: Integrate simplified expression. ? STEP 4: Write answer in terms of . ? Using substitutions involving implicit differentiation When a root is involved, it makes thing much tidier if we use Q Use the substitution to find STEP 1: Using substitution, work out and (or variant)

=2 = 2 5 ? = 2 2 STEP 2: Substitute these into expression. ? STEP 3: Integrate simplified expression. ? STEP 4: Write answer in terms of . ? This was marginally less tedious than when we used , as we didnt have fractional powers to deal with. Test Your Understanding

Edexcel C4 Jan 2012 Q6c Hint: You might want to use your double angle formula first. (As before is going to be messy) ? Definite Integration Now consider: Q Calculate ? Use substitution: ? Now because weve changed from to , we have to work out what values of would have given those limits for : When , When ? ? Test Your Understanding Edexcel C4 June 2011 Q4

When , When ? Exercise 6F Use the given substitution to integrate. Use an appropriate substitution. 1 a ? c c ? e ? 3 a ? ?

e ? 2 a ? c ? e ? SKILL #7: Integration by Parts cos =? Just as the Product Rule was used to differentiate the product of two expressions, we can often use Integration by Parts to integrate a product. ! To integrate by parts: The Product Rule: On the right-hand-side, both and are the product of two expressions. So if we made either the subject, we could use say to represent in the example. Rearranging: Proof ? (not needed for exam)

Integrating both sides with respect to , we get the desired formula. SKILL #7: Integration by Parts cos =? ? = STEP 1: Decide which thing will be (and which ). Youre about to differentiate and integrate , so the idea is to pick them so differentiating makes it simpler, and can be integrated easily. will always be the term UNLESS one term is . ? STEP 2: Find and . STEP 3: Use the formula. ? I just remember it as minus the integral of the two new things timesed together

Another Example Q Find This time, we choose to be the because it differentiates nicely. ? STEP 1: Decide which thing will be (and which ). STEP 2: Find and . STEP 3: Use the formula. IBP twice! Q Find = = 2 We have to apply IBP again! ?

Therefore Bro Tip: I tend to write out my working for any second integral completely separately, and then put the result back into the original integral later. Test Your Understanding Q Find 2 sin 2 cos +2 cos + ? Integrating and definite integration Q Find , leaving your answer in terms of natural logarithms. = ln ? =1 Q Find , leaving your answer in terms of natural logarithms. If we were doing it from scratch: ?

In general: Test Your Understanding Q Find = =sin ? Exercise 6G 1 a c ? c e ? 3 a ? ?

e ? 4 a ? ? ? c 2 a ? (hint: ) ? ? e ?

Bonus Question: g ? c e ? You will need the following standard results (given in your formula booklet). Well prove them later. ! SKILL #8: Integrating top-heavy algebraic fractions 2 =? +1 How would we deal with this? (the clues in the title) +1

1 2 + 0 +0 2 to simplify Some manipulation + ? +0 1 Now integrate ? Test Your Understanding 1? ? Trapezium Rule Revisited Trapezium Rule: You already know the Trapezium Rule from C2. You may be asked similar questions in a C4 exam, except now the expressions involve integrations rules youve just learnt.

Given a) Find the exact value of . Q b) Use the trapezium rule with two strips to estimate . c) Use the trapezium rule with four strips to find a second estimate of . d) Find the percentage error in using each estimate. a =[ ln|sec ?+tan |] 3 0 c b ? ? 1 [

1+ 2 (1.155 ) +2 ] =1.39 2 6 1 [ 1+2 ( 1.035+1.155+ 1.414 ) + 2 ]=1.34 2 12 d Error is 5.6% and 1.5%. ? Test Your Understanding a) b) Q c) d) Find the exact value of . Use the trapezium rule with 4 strips to estimate . Use the trapezium rule with 6 strips to find a second estimate of . Compare the percentage error in using each estimate. a ? 1.34

b 1.42 c ? ? 11.4% and 61.% d ? SKILL #9: Volumes of Revolution gives the area bounded between , , and the -axis. Why? If we split up the area into thin rectangular strips, each with width and each with height the for that particular value of . Each has area . If we had discrete strips, the total area would be: But because the strips are infinitely small and we have to think continuously, we use instead of . Integration therefore can be thought of as a continuous version of summation.

1 2 SKILL #9: Volumes of Revolution Now suppose we spun the line about the axis to form a solid (known as a volume of revolution): Click to Start Bromanimation How could we use the same principle as before to express the volume? What should we use instead of strips?

Reveal >> Were summing a bunch of infinitely thin cylinders, each of width and radius . Each has a volume of: ? Thus the volume is: ! i.e. Square the function and slap a on front. ? Examples The region is bounded by the curve with equation , the -axis and the lines and . Q a) Find the area of . b) Find the volume of the solid formed when the region is rotated through radians about the -axis. Q a ?

b ? Test Your Understanding: Do the same when , bound between and . a ? b ? Area/Volume with Parametric Equations I once again have parametric equations: I want the integration to be in terms of rather than . Area: Volume: ? ? Bro Memory Tip: No need to remember the whole new formulae. All you need to remember is that , which is obvious since

the s cancel. Bro Tip: Remember that you have to change the bounds! Example: Find the volume of revolution and the area bound by the curve and and . =2 ? ? ? STEP 1: Find STEP 2: Change limits STEP 3: Integrate Test Your Understanding June 2011 Q7 When , When , ? Exercise 6I

1 Find the (i) area of region and (ii) solid of revolution bound by lines and for: ? a c (Hint: use integration by parts with for ) ? ? ? ? lines and . The ? parametric e The region is bound by the curve , the -axis and the equations for are and . Find: a) The area of . 3 b) The volume of the solid of revolution formed about the -axis. ? equations and The curve has parametric c) Find the area of the region bounded by and the -axis. d) Find the volume of the solid of revolution. ?

4 ? ? SKILL #10: Differential Equations (Were on the home straight!) Earlier in C4 we encountered differential equations. These are equations involving a mix of variables and derivatives, e.g. , and . Solving these equations means to get in terms of (with no ). Q Find the general solution to STEP 1: Get to the side of by dividing and to the other side. = ( +1 ) ? (you may need to factorise) ? STEP 2: Multiply both sides by , then integrate both sides.

? STEP 3: Make the subject, if the question asks. Another Example Q Find the general solution to ? ? ? ? STEP 1: Get to the side of by dividing and to the other side. (you may need to factorise) STEP 2: Multiply both sides by , then integrate both sides. STEP 2b: If possible, try to combine your constant of integration with other terms (e.g. by letting where is another constant) STEP 3: Make the subject, if the question asks.

Test Your Understanding The rate of increase of a rabbit population (with population , where time is Q ) is proportional to the current population. Form a differential equation, and find its general solution. ? (Notice by the way that , and since is a constant, we could always write . i.e. The general solution is any generic exponential function, not just restricted to those with as the base. However it is customary to write ) Differential Equations with Boundary Conditions Q Find the general solution to Given that when . Leave your answer in the form Use partial fractions to split up RHS. When : ? Bro Tip: While it doesnt matter when you determine your constant using the boundary conditions,

usually its easiest to determine it as soon as possible. Key Points Get on to LHS by dividing (possibly factorising first). If after integrating you have on the RHS, make your constant of integration . Be sure to combine all your s together just as you did in C2. E.g.: ? Sub in boundary conditions to work out your constant better to do sooner rather than later. Exam questions partial fractions. Exercise 6J 1 a Find the general solution in the form . ? ? c e 2

a c e 4 a c e Find the general solution (no need to put in form ) ? ? Find particular solutions using boundary conditions. ? ? ? ? ? Exercise 6K 1 The size of a certain population at time is given by . The rate of increase of I given by . Given that a time , the population was 3, find the population at time . The mass at time ? of the leaves of a certain plant varies according to: 3 a) Given that at time , find an expression for in terms of .

b) Find a value for when . c) Explain what happens to the value of as a increases ? approaches 1. ? , where is measured in hours. The thickness of ice mm on a pond is increasing and ? Find how long it takes the thickness of ice to increase from 1mm to 2mm. 5 The rate of increase of the radius km of an oil slick is given by , where is a positive constant. When the slick was first observed the radius was 3km. Two days later it was ? 5km. Find, to the nearest day when the radius will be 6. 7 ? Summary of Functions How to deal with it Standard Standard result result Standard result In

formularesult booklet, but use which is Standard of the form For both and use identities for ? ? ? Would use substitution , but too hard for exam. Would use substitution , but too hard for exam. which is of the form ? (+constant) Formula booklet? No No No Yes No No Yes

No No Yes No Yes Yes No ? ? No Yes ? ? Yes Yes ? Summary of Functions How to deal with it By observation. By observation. (+constant)

? ? ? For any product of sin and cos with same coefficient of , use double angle. ? Use IBP, where Use algebraic division. Use partial fractions. ? ? ? Use partial fractions. ? Formula booklet? No! Yes (but memorise) No No

No No No No Summary of Functions How to deal with it (+constant) Reverse chain rule. Of form Power around denominator so NOT of ? form . Rewrite as product. Reverse chain rule (i.e. Consider and differentiate) ? inner function For any function where is linear expression, divide by coefficient of Use sensible substitution. or even better, . Reverse chain rule. ? ? Reverse chain rule.

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