Integrals of the form

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 , than . Now, we have:

Try yourself

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 , than . Now, we have:

Try yourself

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.

Try yourself