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# 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:**

**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 2**

**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

When x = 2, y = -2,

Therefore:

4 − 10 + c = -2

Therefore:

c = 4, and the equation of the graph is y = x^{2} − 5x + 4