Question

Evaluate \[ \int \frac{x + 1}{(x - 2) \sqrt{1 - x}} \, dx = ? \]

• A. \(\log(x + 1) - \log (x - 2) \sqrt{1 - x} + c\)
• B. \(\log(x - 2) \sqrt{1 - x} + c\)
• C. \(6 \tan^{-1} \sqrt{1 - x} - 2 \sqrt{1 - x} + c\)
• D. \(4 \tan^{-1} \sqrt{1 - x} - 2 \sqrt{1 - x} + c\)

Hint:

Use substitution and standard integral forms involving inverse trigonometric functions.

Answer 1

The Correct Option is C

Use substitution \(t = \sqrt{1 - x}\) to simplify integral. Integrate using inverse tangent and algebraic terms to get the solution \[ 6 \tan^{-1} \sqrt{1 - x} - 2 \sqrt{1 - x} + c. \]