Integration by Parts »

The substitution method turns an unfamiliar integral into one we can evaluate. In other words, substitution gives us a simpler integral involving the variable u. This lesson shows how the substitution technique works.

Let's now review the five steps for integration by substitution.

Step 1: Choose a new variable u

Step 2: Determine the value dx

Step 3: Make the substitution

Step 4: Integrate resulting integral

Step 5: Return to the initial variable x

Integration by substituting u = ax + b

These are typical examples where we use the method of subsitution.

Example 1: Evaluate

 

Solution:

Step 1: Chose a substitution function u

The substitution function is  

Step 2: Determine the value dx

 

Step 3: Make the substitution

 

Step 4: Integrate resulting integral

 

Step 5: Return to the initial variable: x

 

So, the solution is:

Example 2: Evaluate

 

Solution:

Step 1: Chose a substitution function u

The substitution function is  

Step 2: Determine the value dx

 

Step 3: Make the substitution

 

Step 4: Integrate resulting integral

 

Step 5: Return to the initial variable: x

 

The solution is:

Exercise 1: Evaluate using substitution u = ax + b

More complicated examples

The steps for doing integration by substitution in this section are the same as the steps for previosu one, but we have to chose our substitution function wisely.

Example 3: Find

 

Solution:

 

Example 4: Find

 

Solution:

 

Example 5: Find

 

Solution:

 

Exercise 2: Evaluate using substitution