“Teach A Level Maths” Vol. 2: A2 Core Modules 23: Integrating some Compound Functions © Christine Crisp

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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.

SUMMARY The rule for integrating a compound function ( a function of a function ) is: integrate the outer function divide by the derivative of the inner function provided that the inner function is linear Add C There is NO single rule for integration if the inner function is non-linear.

Exercises Without working them out, decide which of the following can be integrated using the rule we have found in this section. 1. 2. 3. 4. 5. 6. 7. 8. We’ll now practise some integrals like 1, 2, 3, 6 and 8.

e.g. 1. e.g. 2. Integrate the outer function

e.g. 1. e.g. 2. Integrate the outer function

e.g. 1. e.g. 2. Integrate the outer function Divide by the derivative of the inner function -3. If we write we have a clumsy “piled up” fraction so we put the (-3 ) beside the 5.

e.g. 1. e.g. 2. Integrate the outer function Divide by the derivative of the inner function -3.

e.g. 3. This is related to Integrate the outer function:

e.g. 3. This is related to Integrate the outer function: Divide by the derivative of the inner function

e.g. 3. This is related to Integrate the outer function: Divide by the derivative of the inner function

e.g. 3. This is related to Integrate the outer function: Divide by the derivative of the inner function

Exercises Find 1. 2. 3. 4. 5. Solutions: 1. 2.

3. 4. 5.

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.

The rule for integrating a compound function ( a function of a function ) is: SUMMARY integrate the outer function divide by the derivative of the inner function provided that the inner function is linear There is NO single rule for integration if the inner function is non-linear. Add C

Only the functions marked with a tick can be integrated by the rule. 1. 2. 3. 4. 5. 6. 7. 8.

Integrate the outer function Divide by the derivative of the inner function -3. e.g. 2. ( Avoiding “piled up” fractions )

e.g. 3. This is related to So,