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**“Teach A Level Maths” Vol. 2: A2 Core Modules**

23a: Integrating (ax+b)n © 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 e.g. For we get

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**Since indefinite integration is the reverse of differentiation, we get**

So, The rule is: integrate the outer function

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**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

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**Since indefinite integration is the reverse of differentiation, we get**

So, The rule is: integrate the outer function If we write divide by the derivative of the inner function we have a clumsy “piled up” fraction so we put the 2 beside the 5.

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**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.

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The rule is: integrate the outer function divide by the derivative of the inner function i.e. the coefficient of x Tip: We can check the answer by differentiating it. We should get the function we wanted to integrate. Make power one more Drop it through the trap door Divide by coefficient of x

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**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

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**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.

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**? What is the important difference between and**

When we differentiate the inner function of the 1st example, we get 2, a constant. Dividing by the 2 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.

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SUMMARY The rule for integrating a compound function ( a function of a function ) is: integrate the outer function divide by 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.

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e.g. 1. Integrate the outer function Divide by derivative of the inner function i.e. the coefficient of x ( Remembering not to pile up the fractions )

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2. 4.

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