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- Integration by Substitution

Integration Techniques: (lesson 1 of 4)

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**

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:

Level 1

Level 2

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:

Level 1

Level 2