May 27, 2021

Chapters

## Integration Notation

As a reminder, integration** notation** is important to understand. There are two main integrals you will encounter:

A | B | C | |

Indefinite | Integral without bounds | Function | Variable to be integrated |

Definite | Integral with upper bound m and lower bound n | Function | Variable to be integrated |

In the table above, you can see that the only **difference between** indefinite and definite variables is that definite variables specify the interval for which the area needs to be calculated.

A | B | C |

To get the area between an interval, you simply need to take the area of the upper limit **minus** the area of the lower limit.

Step 1 | Find the integral of | |

Step 2 | Get the area of the upper limit | = 5 |

Step 3 | Get the area of the lower limit | = 2 |

Step 4 | Get the area between the limits | 5-2 = 3 |

## Integration Rules

You can think of integration as the **opposite** as taking the **derivative.** Let’s compare each by seeing the rule for taking the integral and derivative or a power.

Start with a power function | f = |

Take the derivative | f’ = = |

Take the integral of the derivative | = = = = |

In the table above, we can see that by taking the **integral** of the derivative, we have come back to the original function. Take a look at more integration rules and examples below.

Integral Function | Result | Example | |

Constant | = 2x + c | ||

Power | = = | ||

Fraction of x | = | ||

Natural Log | = | ||

Exponential | = | ||

Log | = | ||

Sum of two functions | + | = + | |

Two functions subtracted | - | = - |

## Integration by Parts

When you have two functions **multiplied** by each other, you can use integration by parts. The rule is as follows:

Element | Description |

u(x), u | The first function |

v(x), v | The second function |

u’(x), u’ | The derivative of the first function |

Say you have the following function to **integrate:**

We can use the **rule** in order to find the integral.

Element | Description | Result |

u(x), u | The first function | x |

v(x), v | The second function | |

u’(x), u’ | The derivative of the first function | 1 |

**Plugging** this into the rule, we get:

Now we **solve** for each term:

A | |

B | |

C | = |

The final **result** is:

## Integration by Substitution

Also called u-substitution, Integration by **substitution** can be used if you have two functions, one of which can be written as the derivative of the first function.

The goal is to simplify the integration process. So, we can pick the **inside** function as our ‘u.’

Now that we selected a u and have attained the **derivative,** what we need to do now is get the dx term by itself.

Now, we simply replace the function that we’ve chosen as our u by the u term and **replace** the dx term with what we just solved.

Now, we **simplify.**

Recall that a function multiplied by a **constant** in an integral is just that constant multiplied by the integral.

Now, instead of the complicated formula we had in the **beginning,** we can solve the integral easily.

The final step is to simply **substitute** the ‘u’ back into the equation.

## Problem 1

Integrate the following function using any of the **rules** above:

## Problem 2

Integrate the **following** function using any of the rules above:

## Problem 3

Integrate the following functions using integration by **parts:**

## Problem 4

Integrate the following function using **u-substitution.**

## Problem 5

Integrate the following functions using any of the **rules** above:

## Solution Problem 1

Let’s **integrate** this function using u-substitution:

## Solution Problem 2

Let’s **integrate** the function:

## Solution Problem 3

For this **function** we use integration by parts:

The integration of is exactly the **same** as in problem 1.

Let’s focus on the **second** term.

The integration of is the **same** as problem 1.

## Solution Problem 4

First, you need to **pick** which function will be your u term.

Next, take the derivative and **solve** for dx.

Plug **everything** back into the equation and simplify.

Lastly, plug the **u** term back in.

## Solution Problem 5

In this integral, you first need to use the **difference** rule.

Use integration by parts for the **first** integral.

Use u-substitution for the **second** integral.

Finally, put it all **together.**