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Integration Techniques: (lesson 1 of 4)

Integration by Substitution

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

Integration by Substitution example

Solution:

Step 1: Chose a substitution function u

The substitution function is substitution function

Step 2: Determine the value dx

Compute du

Step 3: Make the substitution

Make the substitution

Step 4: Integrate resulting integral

Integrate resulting integral

Step 5: Return to the initial variable: x

Substitute back

So, the solution is: result

Example 2: Evaluate

Integration by Substitution example

Solution:

Step 1: Chose a substitution function u

The substitution function is substitution function

Step 2: Determine the value dx

Compute du

Step 3: Make the substitution

Make the substitution

Step 4: Integrate resulting integral

Integrate resulting integral

Step 5: Return to the initial variable: x

Substitute back

The solution is: result


Exercise 1: Evaluate using substitution u = ax + b

Level 1

Exercise 1 answer 1
answer 2
answer 3
answer 4

Level 2

Exercise 2 answer 1
answer 2
answer 3
answer 4

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

Integration by Substitution example

Solution:

Integration by Substitution solution

Example 4: Find

Integration by Substitution example

Solution:

Integration by Substitution solution

Example 5: Find

Integration by Substitution example

Solution:

Integration by Substitution solution

Exercise 2: Evaluate using substitution

Level 1

Exercise 1 answer 1
answer 2
answer 3
answer 4

Level 2

Exercise 2 answer 1
answer 2
answer 3
answer 4