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« Integration by Parts
Integration Techniques: (lesson 3 of 4)

Integrals Involving Trigonometric Functions

Integrals of the form Trigonometric Integrals Form

Case 1: m is an odd integer :

Step 1: Write Step 1.

Step 2: Apply identity: Step 2

Step 3: Use the substitution Step 3.

Example 1: Evaluate the following integral

Example 1

Solution:

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

Step 1:

Solution Step 1

Step 2: Apply identity: Step 2. We continue

Solution Step 2

Step 3: Use the substitution Step 3.

If Solution Step 3, then Solution Step 3. Now, we have:

Solution Step 3

Try yourself

exercisie
answer 1 answer 2 answer 3 answer 4

Case 2: n is an odd integer :

Step 1: Write Step 1.

Step 2: Apply identity: Step 2

Step 3: Use the substitution Step 3.

Example 2: Evaluate the following integrl

Example 1

Solution:

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

Step 1:

Example 1

Step 2: Apply identity: Step 2.

Solution Step 2

Step 3: Use the substitution Step 3.

If Solution Step 3, then Solution Step 3. Now, we have:

Solution Step 3

Try yourself

exercisie
answer 1 answer 2 answer 3 answer 4

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

Trigonometric Integrals Case 3

Example 3: Evaluate the following integrl

Example 1

Solution:

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

Solution

Try yourself

exercisie
answer 1 answer 2 answer 3 answer 4