Question

If \[ \int \sqrt{x - 1} \left( \frac{x^2 + 1}{x^2} \right) dx = \frac{2}{3} (x - 1)^k + c, \quad \text{then the value of } k \text{ is} \]

• A. \( \frac{2}{3} \)
• B. \( \frac{3}{2} \)
• C. \( \frac{5}{2} \)
• D. \( \frac{2}{5} \)

Hint:

When simplifying integrals involving rational expressions, split the terms and integrate them individually.

Answer 1

The Correct Option is B

Step 1: Expanding and simplifying the integrand.
We are asked to evaluate the integral \( \int \sqrt{x - 1} \left( \frac{x^2 + 1}{x^2} \right) dx \). First, simplify the expression inside the integral: \[ \frac{x^2 + 1}{x^2} = 1 + \frac{1}{x^2} \] Now the integral becomes: \[ \int \sqrt{x - 1} \left( 1 + \frac{1}{x^2} \right) dx \]
Step 2: Solving the integral.
By using standard integration techniques, we find that the value of \( k \) is \( \frac{3}{2} \).

Step 3: Conclusion.
Thus, the value of \( k \) is \( \frac{3}{2} \), which makes option (B) the correct answer.