May 27, 2021

Chapters

## What is Integration?

Integration is an **operation** which helps you determine the area under the graph of a function.

A | B | C | |

Integral | Integral sign | Function | Tells us which variable we’re integrating |

Example | 2b + 4 | db |

Notice that we don’t always have to use ‘dx,’ as this part of an integration **formula** will depend on which variable we’re integrating.

When we plot the example from the table above, the integration of this function can give us the** area** underneath it.

## Indefinite Integral

An indefinite integral is an integral whose **bounds** are limited to the graph of the function.

A | B | C | |

Integral | Integral sign which has no upper or lower bound | Function | Tells us which variable we’re integrating |

As you can see, an indefinite integral has no **upper** or **lower** bound. This means that this integrated formula can give us the area under any point on the graph:

## Definite Integral

A definite integral has bounds that are** defined** in the integration equation.

A | B | C | |

Integral | Integral sign which has upper bound of m and lower bound of n | Function | Tells us which variable we’re integrating |

As you can see, the main difference between an **indefinite** and a definite integral is that a definite integral is defined by bounds of the interval [n,m].

In order to get the area of the** interval** between n and m, we simply do the following:

Step 1 | Integrate the function | = |

Step 2 | Plug in m to the integrated equation | = 30 |

Step 3 | Plug n to the integrated equation | = 6 |

Step 4 | Get m - n | 30 - 6 = 24 |

## Integration Rules

Now that you know the **logic** behind what integration is and why we use it, take a look at integration rules.

### Constant

Integral Function | Rule | |

Constant | ax |

Here are some **examples**:

Integral Function | Result | |

1 | 4x + c | |

2 | x + c | |

3 | 2x + c |

Integration can be thought of doing the **opposite** of finding the derivative. Take a look at the example below.

Function | Derivative | Integral |

4x + 2 | = | |

f = 4x + 2 | f’ =4 | = 4s + c |

We can never get the exact constant of the **original formula**, but we can put ‘c’ there to stand in for the constant that was there.

### Fraction

There are two ways you can integrate a fraction. First, **convert** the fraction into a regular number.

Instead of integrating the fraction, we can use common **exponent** rules to turn it into two numbers multiplied by each other.

Another way you can integrate a fraction is to see if it follows the rule for **integrating** reciprocals.

You can combine any of the methods we discussed, as well as use them with integration by parts and **u-substitution**.

### Sum

The sum rule of integration is a **simple** rule. It says that if you have to integrate the sum of two functions, the result is equal to the integration of both functions added together.

What | Example | |

f(x) | First function | x+3 |

g(x) | Second function | 5x |

Let’s take a look at an **example**.

First, we simplify the **expression**.

Next, because it is the addition of two **functions**, we can integrate them separately.

### Difference

The difference rule of integration is similar to the** sum** rule. It says that if you have to integrate the difference of two functions, the result is equal to the integration of one function minus the other.

What | Example | |

f(x) | First function | 2x |

g(x) | Second function (that we subtract from the first) | 4x+3 |

Let’s take a look at an **example**.

First, we simplify the **expression**.

Next, because it is the **difference** of two functions, we can integrate them separately.

### Power

Powers are a very important part of integration. The **rule** is as follows.

x | The variable we’re integrating |

n | The exponent, or power |

What about when you don’t have any **power**? Take a look at this example.

Well, in this case, the power is actually there. It is equal to one, because a **number** to the power of 1 is just itself. So, it would be the following.

## Integration by Parts

Integration by parts is exactly as it sounds. When you have two functions** multiplied** together, you can use integration by parts to integrate them.

What | Example | |

u | The first function | 3x |

v | The second function | |

u’ | The derivative of the first function | 3 |

Keep in mind, however, that you should only integrate by parts if you have two functions that will not produce more **functions** that need to be integrated by parts. To understand this, let’s take an example that we actually can use this method for.

Now, we follow the rule.

Simplifying only the **highlighted** term, we get the following.

Now, imagine if we didn’t have this term, but** instead** had this.

When we simplify this term, we end up with another two functions multiplied by one another. This means that we would need to keep integrating by parts infinitely. If you end up with this, it is better to **integrate** by u-substitution.

## Integration by Substitution

Integration by substitution, also called u-substitution, is an easy way to integrate. Let’s take a look at the** notation** and a simple example.

u | The expression we want to replace | 4x |

du | The derivative of the expression | 4 dx |

dx | Solve for dx |

Continuing with our example from above, let’s take a look at the **steps** in u-substitution.

1 | Replace the u and dx values in the formula | = |

2 | Simplify the expression | = |

3 | Integrate like normal | |

4 | Simplify | |

5 | Plug the original value of u back in to get the final result | = |