Question

Evaluate \[ \int \frac{dx}{(1+\sqrt{x})\sqrt{x-x^2}}. \]

• A. \[ -2\sqrt{\frac{1+\sqrt{x}}{1-\sqrt{x}}}+C \]
• B. \[ -\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}+C \]
• C. \[ -2\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}+C \]
• D. \[ 2\sqrt{\frac{1+\sqrt{x}}{1-\sqrt{x}}}+C \]

Hint:

For integrals containing \(\sqrt{x-x^2}\), the substitution \(\sqrt{x}=\sin\theta\) is usually effective because \[ x-x^2=\sin^2\theta\cos^2\theta. \] This converts the radical into a simple trigonometric product.

Answer 1

The Correct Option is C