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

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

## Example

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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