Question

Evaluate the integral \[ \int \frac{dx}{\sqrt{(x - 1)(x - 2)}} = \text{?}

• A. \( \log \left( \left| x - \frac{3}{2} \right| \sqrt{x^2 - 3x + 2} \right) + c \)
• B. \( \log \left( \left| x - \frac{3}{2} \right| + \sqrt{x^2 - 3x + 2} \right) + c \)
• C. \( \log \left( |x - 1| + \sqrt{x^2 - 3x + 2} \right) + c \)
• D. \( \log \left( \left| x + \frac{3}{2} \right| \sqrt{x^2 - 3x + 2} \right) + c \)

Hint:

For integrals involving square roots of quadratic expressions, use substitution and known formulas to simplify the integration.

Answer 1

The Correct Option is B

Step 1: Simplify the integrand.
We are asked to integrate \( \frac{dx}{\sqrt{(x - 1)(x - 2)}} \). To handle this, first simplify the denominator and make a substitution to simplify the integral.
Step 2: Use the standard integral.
This is a standard integral and can be integrated using the formula for the integral of rational functions with square roots. After applying the formula, we get: \[ \log \left( \left| x - \frac{3}{2} \right| + \sqrt{x^2 - 3x + 2} \right) + c \]
Step 3: Conclusion.
Thus, the value of the integral is \( \log \left( \left| x - \frac{3}{2} \right| + \sqrt{x^2 - 3x + 2} \right) + c \), corresponding to option (B).