“Teach A Level Maths” Vol. 2: A2 Core Modules 24: Differentiation and Integration with Trig Functions © Christine Crisp

Module C4 OCR This presentation follows on from “Differentiating some Trig Functions” "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"

In C3 you learnt to differentiate using the following methods: the product rule the quotient rule You also learnt to integrate some compound functions using : inspection This presentation gives you some practice using the above methods with trig functions.

Reminder: A function such as is a product, BUT we don’t need the product rule. When we differentiate, a constant factor just “tags along” multiplying the answer to the 2nd factor. However, the product rule will work even though you shouldn’t use it N.B.

Reminder: A function such as is a product, BUT we don’t need the product rule. When we differentiate, a constant factor just “tags along” multiplying the answer to the 2nd factor. However, the product rule will work even though you shouldn’t use it so, as before.

Exercise Use the product rule to differentiate the following. 1. 2.

Solutions: 1. Let and Remove common factors:

2. Let and

For products we use the product rule and for functions of a function we use the chain rule. Decide how you would differentiate each of the following ( but don’t do them ): (a) (b) (c) (d) Chain rule Product rule This is a simple function Chain rule

Exercise Decide with a partner how you would differentiate the following ( then do them if you need the practice ): 1. 2. 3. Write C for the Chain rule and P for the Product Rule P C C or P

Solutions 1. P 2. C

3. C Either P Or

The quotient rule is Use the quotient rule to differentiate the following. 1. 2. Exercise

Solution: 1. and

2. Solution: and

We can now differentiate the trig function by writing x y tan =

So, This answer can be simplified: is defined as Also, So,

Exercise Use the quotient rule ( or, for (a) and (b), the chain rule ) to find the derivatives with respect to x of Before you check the solutions, look in your formula books to see the forms used for the answers. Try to get your answers into these forms.

(a) Solution:

(b) Solution:

(c) Solution:

Since integration is the reverse of differentiation, for the trig ratios we have

e.g. 1 Radians! The definite integral can give an area, so this result may seem surprising. However, the graph shows us why it is correct. The areas above and below the axis are equal, . . . This part gives a positive integral but the integral for the area below is negative. This part gives a negative integral

e.g. 1 Radians! The definite integral can give an area, so this result may seem surprising. However, the graph shows us why it is correct. How would you find the area? This part gives a positive integral Ans: Find the integral from 0 to and double it. This part gives a negative integral

2. N.B. The area is above the axis, so the integral gives the entire area.

Before we try to integrate compound functions, we need to be able to recognise them, and know the rule for differentiating them. where , the inner function. If We saw that in words this says: differentiate the inner function multiply by the derivative of the outer function we get e.g. For

Since indefinite integration is the reverse of differentiation, we get 3 3 So, If we divide C by 3, we get another constant, say C1, but we usually just write C. The rule is: integrate the outer function

Since indefinite integration is the reverse of differentiation, we get So, The rule is: integrate the outer function divide by the derivative of the inner function

Since indefinite integration is the reverse of differentiation, we get So, The rule is: integrate the outer function divide by the derivative of the inner function

Since indefinite integration is the reverse of differentiation, we get So, The rule is: integrate the outer function divide by the derivative of the inner function Tip: We can check the answer by differentiating it. We should get the function we wanted to integrate.

However, we can’t integrate all compound functions in this way. Let’s try the rule on another example: THIS IS WRONG ! e.g. integrate the outer function divide by the derivative of the inner function

However, we can’t integrate all compound functions in this way. Let’s try the rule on another example: THIS IS WRONG ! e.g. The rule has given us a quotient, which, if we differentiate it, gives: . . . nothing like the function we wanted to integrate.

? What is the important difference between and When we differentiate the inner function of the 1st example, we get 3, a constant. Dividing by the 3 does NOT give a quotient of the form ( since v is a function of x ). The 2nd example gives 2x,which is a function of x.

? What is the important difference between and When we differentiate the inner function of the 1st example, we get 3, a constant. Dividing by the 3 does NOT give a quotient of the form ( since v is a function of x ). The 2nd example gives 2x,which is a function of x. So, the important difference is that the 1st example has an inner function that is linear; it differentiates to a constant. The 2nd is non-linear. We can’t integrate it.

Exercises 1. Find 2. Solutions: 1. 2.

We found earlier that so, You need to remember this exception to the general rule that we can only integrate directly if the inner function is linear. We also have, for example,

There are 3 more important trig integrals. e.g. Find We have a function of a function . . . but the inner function . . .

There are 3 more important trig integrals. e.g. Find We have a function of a function . . . but the inner function . . . is not linear. However, we can use a trig formula to convert the function into one that we can integrate.

e.g. Find Which double angle formula can we use to change the function so that it can be integrated? ANS: Rearranging the formula: So,

The previous example is an important application of a double angle formula. The next 2 are also important. Try them yourself. 1. Find 2. Find Exercise

Exercise 1. Find Solution: So,

2. Find Solution: So,

SUMMARY The rearrangements of the double angle formulae for are They are important in integration so you should either memorise them or be able to obtain them very quickly.

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.

SUMMARY Otherwise use the product rule: If where u and v are both functions of x To differentiate a product: Check if it is possible to multiply out. If so, do it and differentiate each term.

For products we use the product rule and for functions of a function we use the chain rule. (b) (c) (d) Product rule Chain rule This is a simple function For example,

The quotient rule is

With the quotient rule we can differentiate the trig function by writing x y tan =

So, This answer can be simplified: is defined as Also,

Since integration is the reverse of differentiation, for the trig ratios we have

e.g. 1 Radians! The definite integral can give an area, so this result may seem surprising. However, the graph shows us why it is correct. This part gives a negative integral This part gives a positive integral To find the area, find the integral from 0 to and double it.

Before we try to integrate compound functions, we need to be able to recognise them, and know the rule for differentiating them. where , the inner function. If e.g. For We saw that in words this says: differentiate the inner function multiply by the derivative of the outer function we get

Since indefinite integration is the reverse of differentiation, we get integrate the outer function divide by the derivative of the inner function So, The rule is: Tip: We can check the answer by differentiating it. We should get the function we wanted to integrate.

? What is the important difference between and When we differentiate the inner function of the 1st example, we get 3, a constant. The 2nd example gives 2x,which is a function of x. Dividing by the 3 does NOT give a quotient of the form ( since v is a function of x ). So, the important difference is that the 1st example has an inner function that is linear; it differentiates to a constant. The 2nd is non-linear. We can’t integrate it. What is the important difference between and ?

We found earlier that so, You need to remember this exception to the general rule that we can only integrate directly if the inner function is linear. We also have, for example,

e.g. Find We have a function of a function . . . but the inner function is not linear However, we can use a trig formula to covert the function into one that we can integrate. There are 3 more important trig integrals.

Which double angle formula can we use to change the function so that it can be integrated? e.g. Find ANS: Rearranging the formula: So,

SUMMARY The rearrangements of the double angle formulae for are They are important in integration so you should either memorise them or be able to obtain them very quickly.