May 27, 2021

Chapters

## Integration

Integration is an important **concept** of calculus. When you integrate an equation, you are simply finding the area beneath that equation’s graph. To understand better, take a **linear** equation:

We can easily find the area beneath this line by finding the **distance** between the line and the axis, or even by simply counting the squares underneath it. Take a look at another example:

While we can count the area with whole squares, how can we **measure** the other squares? Do we take only half, or a third? This is where integration comes in:

## Integration Notation

When we’re interested in integrating a function, this is also known as the **reverse** of the derivative.

A | B | C |

Function | First derivative of the function | Integral of the function |

Take a look at the **notation** of integration:

A | Integration symbol |

B | Function |

C | This tells you which variable to integrate |

An integral can be either **definite** or **indefinite.**

B | A |

Definite | Indefinite |

The area between a defined interval | The area of the entire function |

You will mostly encounter **definite** intervals.

## Constants

Constants are the easiest to integrate. The rule for integrating **constants** can be found in the table below.

Integral | Rule | Example | |

Constant | ax + c | = 4x + c |

The reason why we put a constant c is because the derivative of any **constant** is zero.

Function | Derivative | Integral |

4x + 30 | f’ = 4x + 0 | 4x dx = 4x + c |

4x + 2 | f’ = 4x + 0 | 4x dx = 4x + c |

4x | f’ = 4x | 4x dx = 4x + c |

Adding the constant c to the integral reflects the fact that there might have been a constant in the **original** function.

Let’s take the following as an example:

To find the area, we simply subtract the upper interval and the **lower** interval.

## Exponents

Integrating exponents follows a similar **process** as above. Take a look at the exponent rule below.

This rule should be applied to every **exponent** you want to integrate. Let’s take a look at an example.

The table below has the steps for **integrating** this exponent.

Step 1 | Follow the rule specified above | |

Step 2 | Simplify the fraction |

## Fractions

To integrate fractions, you will need a **mix** of the exponent rule as well as the following rules.

Power/Fraction Rule | = | |

Reciprocal Integral Rule | = ln|x| | |

Integration by Parts | = | |

U-substitution Integration | = |

Let’s take one example and integrate it **two** ways. First, let’s rewrite the function using the power/fraction rule.

Step 1 | Rewrite the function | = |

Step 2 | Integrate using the exponent rule | |

Step 3 | Simplify | = |

Next, we can use u-substitution and the **exponent** rules. First, we write down our parameters.

u | = |

du | |

dx | |

x | x^{-2}\sqrt{x} = \sqrt{\frac{1}{u}}x = \sqrt{\frac{1}{u}} = (u^{-1})^{\frac{1}{2}}x = u^{\frac{1}{2}} |

Next, we replace the function with our **parameters** above.

We also have to **replace** the x in our function.

Simplify the u **terms.**

Now, integrate the **function.**

## Exponential and Logarithms

In order to integrate **exponential** or **logarithmic** function, you will need to follow the rules for integration.

### Exponential Function

Recall that an exponential function is one that has the variable in the **exponent.** Take a look at the notation and an example below.

Notation | Example |

In order to integrate this function, we need to follow the following **rule.**

Integral Exponential | Integral Rule | Example |

### Euler's number

Euler's has a fixed value, which is e **approximately** equal to 2.718. Take a look at the notation and an example below.

Notation | Example |

In order to integrate this function, we **need** to follow the following rule.

Integral Euler's | Integral Rule | Example |

= |

### Natural Logarithm

Natural logarithms have **Euler’s** number as the base. Take a look at the notation and an example below.

Notation | Example |

In order to integrate this **function,** we need to follow the following rule.

Integral Natural Log | Integral Rule | Example |

## Sum Rule

When you have to integrate the **sum** of two functions, you simply integrate each function separately.

Integral Function | Rule |

## Difference Rule

When you have to integrate the **difference** of two functions, it is the same as when you have the sum of two functions.

Integral Function | Rule |