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Integration : Key Stage 5 (AS-Level) : Mathematics Content Knowledge

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Key Stage 5 (AS-Level)
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1. How confident are you that you know and can explain:

a. integration as the reverse of differentiation?


If we are given \frac{{dy}}{{dx}} = x^2  + 4 then asked to find y in terms of x, the process by which we do this is called integration.

There is an infinite number of functions that when differentiated give x2 + 4. Three examples are below:

y = \frac{{x^3 }}{3} + 4x \Rightarrow \frac{{dy}}{{dx}} = x^2  + 4

y = \frac{{x^3 }}{3} + 4x + 3 \Rightarrow \frac{{dy}}{{dx}} = x^2  + 4

y = \frac{{x^3 }}{3} + 4x - 2 \Rightarrow \frac{{dy}}{{dx}} = x^2  + 4

So if we are given the gradient function and asked to find the original function, unless we are given additional information we cannot find the original function completely.

So if \frac{{dy}}{{dx}} = x^2  + 4 then y = \frac{{x^3 }}{3} + 4x + c where c is the constant of integration.

y = \frac{{x^3 }}{3} + 4x + c is called the integral of y = x^2  + 4 with respect to x and is written: \int {x^2  + 4 \ dx = \frac{{x^3 }}{3} + 4x + c} .

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