Introduction

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Introduction

Integration is the inverse of differentiation.

After differentiating, integration is what you use to get back to where you started.

This concept is called the Fundamental Theorem of Calculus.

In order to integrate it is therefore vital that the principles of differentiation are understood - all we are going to do is the opposite of the differentiation work. (So practice your differentiation skills before starting this topic!)

As integration is the opposite of differentiation we can instantly make some basic rules:

Rules

Note: When integrating and differentiating trigonometric functions we must be working in radians - the rules only work in radians.

As with differentiation, addition and subtractions are integrated separately, and multiples are carried through as these examples show.

Example 1

Example 1

Example 2

Example 2

Example 3

Example 3

Notice that each example ends with a '+ c'. This is a constant that would have disappeared when differentiating, and so we include the '+ c' to remind us that there could have been a constant in the original function.

If we know dy/dx (the gradient), then we can integrate this to get the original function. The only problem is that we will not know the value of the constant, the '+ c'. To find this we will need to know one point on the graph and substitute the x- and y-values into our answer to find the correct function. (See Differential Equations later for more information.)

Example:

Find the equation of the graph that passes through (2, -2) and has gradient

ExampleExample

When x = 2, y = -2,

Therefore:

4 − 10 + c = -2

Therefore:

c = 4, and the equation of the graph is y = x2 − 5x + 4

Graph

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