# High School Calculus/Integration by Substitution

### Integration by Substitution[edit]

There is a theorem that will help you with substitution for integration. It is called **Change of Variables for Definite Integrals**.

what the theorem looks like is this

In order to get **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): \alpha **
you must plug ** a** into the function

**g**and to get you must plug

**into the function**

*b***g**.

The tricky part is trying to identify what you want to make your ** u** to be. Some times substitution will not be enough and you will have to use the rules for integration by parts. That will be covered in a different section

**Ex. 1**

Instead of making this a big polynomial we will just use the substitution method.

Step 1

Identify your *u*

Let

Step 2

Identify

Step 3

Now we plug in our limits of integration to our *u* to find our new limits of integration

When

and when

Now our integration problem looks something like this

Step 4

write your new integration problem

When we plug in our *u* it looks like

Step 5

Evaluate the Integral

As you can see this all simplified fairly nice. Using substitution will be hard, for most people, at first. Once you get the hang of doing this it should come to you faster and faster each time.

I'll give you some other problems to work on as well.

**Ex. 2**

**Ex. 3**

### Solutions[edit]

**Ex. 2**

Let

Then **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \operatorname {d}u = \cos (x) \operatorname {d}x}**

When x = 0

and when

Therefore,

**Ex. 3**

Let

Then

plug in our limits to get new limits

When x = -1

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle (-1)^2 + 4 = 5}**

and when x = 2

Our new integration problem is

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \int_{5}^{8} (u)^\frac {1}{2} \operatorname {d}u}**

Giving us

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle = 26}**