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© Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules 23a: Integrating (ax+b) n

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Integrating (ax+b) n 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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Integrating (ax+b) n Since indefinite integration is the reverse of differentiation, we get integrate the outer function The rule is: So,

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Integrating (ax+b) n Since indefinite integration is the reverse of differentiation, we get integrate the outer function So, The rule is: divide by the derivative of the inner function

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Integrating (ax+b) n divide by the derivative of the inner function Since indefinite integration is the reverse of differentiation, we get integrate the outer function So, The rule is: If we write we have a clumsy “piled up” fraction so we put the 2 beside the 5.

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

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Integrating (ax+b) n divide by the derivative of the inner function i.e. the coefficient of x integrate the outer function The rule is: 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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Integrating (ax+b) n However, we can’t integrate all compound functions in this way. e.g. THIS IS WRONG ! Let’s try the rule on another example: integrate the outer function divide by the derivative of the inner function

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Integrating (ax+b) n However, we can’t integrate all compound functions in this way. e.g. THIS IS WRONG ! Let’s try the rule on another example: 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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Integrating (ax+b) n What is the important difference between and ? When we differentiate the inner function of the 1 st example, we get 2, a constant. The 2 nd example gives 2x,which is a function of x. Dividing by the 2 does NOT give a quotient of the form ( since v is a function of x ). So, the important difference is that the 1 st example has an inner function that is linear; it differentiates to a constant.

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Integrating (ax+b) n The rule for integrating a compound function ( a function of a function ) is: SUMMARY integrate the outer function divide by 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

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

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Integrating (ax+b) n 2. 4.

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Integrating (ax+b) n

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