Integration

Step 1: Simplify the integrand using the identity for . We know that: Substitute this into the integral: This simplifies to:

Step 2: Apply a substitution to simplify the integral. Let: The integral now becomes: At this point, we need a further substitution or simplification method to evaluate the integral, but due to its complexity, solving this exactly involves advanced trigonometric substitutions or numerical methods, which might be outside of simple manual calculation.

Step 1: Break the integral into two parts: Step 2: The second integral is straightforward: Step 3: For the first integral, use integration by parts or refer to standard integral tables for . The general form will involve and the arctangent function. Thus, the answer is a combination of these integrals.

Step 1: Use the substitution , so and . Step 2: From the identity , we have: Step 3: Simplify the integrand: Step 4: Use the substitution , so . This gives: Step 5: Integrate: Thus, the result is:

Reflexivity: To prove is an equivalence relation, we verify reflexivity, symmetry, and transitivity:

Symmetry: For any , , which is a multiple of . Thus, .
If , then for some .

Transitivity: This implies , which is also a multiple of . Thus, . If and , then and for .
Adding these, , which is a multiple of . Thus, . Therefore, is an equivalence relation.

Step 1: Simplify the integrand. Using the identity , we can rewrite the integrand as:

Step 2: Now the integral becomes:

Step 3: Solve the integral using integration by parts. Let: Then: Applying the integration by parts formula , we get:

Step 4: Evaluate the integrals: Thus, we have shown that:

Step 1: Use the substitution , which gives . The limits of integration change accordingly from to .

Step 2: Substitute and simplify the integral: Step 3: Use integration techniques to solve the resulting integral, which involves recognizing the standard form and applying known integral results. After solving the integral, we get: Thus, the required result is proved.

Step 1: Decompose the integrand using partial fractions. First, express the rational function as:

Step 2: Multiply both sides by to find the values of , , and .

Step 3: Solve for , , and by equating coefficients of like powers of .

Step 4: Once the partial fraction decomposition is done, integrate each term separately. After integrating, the result is:

Step 1: We apply integration by parts. Let:
- , so
- , so

Step 2: Using the integration by parts formula: Substitute the values:

Step 3: Simplifying the remaining integral:

Step 4: Factor the result: Thus, the correct answer is (A) .

Step 1: Use integration by parts. Let: - , so - , so

Step 2: Apply the integration by parts formula: Substitute the values:

Step 3: Simplify: