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« Integration by Parts |

Integration Techniques: (lesson 3 of 4)

Case 1: m is an odd integer :

**Step 1:** Write .

**Step 2:** Apply identity:

**Step 3:** Use the substitution .

Example 1: Evaluate the following integral

Solution:

In this example **m = 3** and **n = 2**. Because m is an odd integer we have:

**Step 1:**

**Step 2:** Apply identity: . We continue

**Step 3:** Use the substitution .

If , then . Now, we have:

Case 2: n is an odd integer :

**Step 1:** Write .

**Step 2:** Apply identity:

**Step 3:** Use the substitution .

Example 2: Evaluate the following integrl

Solution:

In this example **m = 6** and **n = 7**. Because n is an odd integer we have:

**Step 1:**

**Step 2:** Apply identity: .

**Step 3:** Use the substitution .

If , then . Now, we have:

Case 3: If both m and n are even, we use identities:

Example 3: Evaluate the following integrl

Solution:

In this example **m = 2** and **n = 2**. Because both m and n are even, we use above identities.