Integral

Step 1: Understanding the Concept:
Similar to the previous question, this follows the standard identity .

Step 2: Key Formula or Approach:
Identify and check if the second term in the bracket is .

Step 3: Detailed Explanation:
1. Let . 2. Differentiate with respect to : 3. Observe that the integrand is exactly . 4. Use the identity:

Step 4: Final Answer
The result of the integral is .

Step 1: Understanding the Concept:
This problem utilizes the special integral form . We need to manipulate the terms inside the parentheses to fit this structure.

Step 2: Key Formula or Approach:
Identify a function such that the integrand becomes . Since is not the direct derivative of , we add and subtract inside the bracket.

Step 3: Detailed Explanation:
1. Rewrite the integral by adding and subtracting : 2. Group the terms: 3. Let . 4. Find the derivative : 5. The integral is now in the form . 6. Applying the rule:

Step 4: Final Answer
The integral is .

Step 1: Understanding the Concept:
This is an integral of a rational function that can be solved by decomposing the fraction into simpler parts using partial fractions.

Step 2: Key Formula or Approach:
Use the standard formula: .

Step 3: Detailed Explanation:
1. Express the fraction as a difference: 2. Integrate each term: 3. Apply the log rule: 4. Combine the logs:

Step 4: Final Answer
The integral is .

Step 1: Understanding the Concept:
To integrate a rational function where the denominator contains a high power of , we can manipulate the expression to create a substitution that involves the derivative of that high power.

Step 2: Key Formula or Approach:
Multiply both the numerator and the denominator by to make the substitution easier.

Step 3: Detailed Explanation:
1. Multiply by : 2. Let . Then , so . Also, . 3. Substitute into the integral: 4. Use partial fractions: . 5. Substitute back :

Step 4: Final Answer
The result of the integral is .