Question

Find the integral of \( \frac{(\sqrt{x+1})^2}{x\sqrt{x + 2x + \sqrt{x}}} \).

• A. \( \sqrt{x} + k \)
• B. \( \frac{1}{2} \sqrt{x} + k \)
• C. \( 2 \sqrt{x} + k \)
• D. \( 2x + k \)

Hint:

For complex integrals, simplify the expression first by combining like terms or using substitution methods.

Answer 1

The Correct Option is C

To integrate the given expression, we use standard integration techniques for rational and square-root functions. The result simplifies to: \[ \int \frac{(\sqrt{x+1})^2}{x\sqrt{x + 2x + \sqrt{x}}} dx = 2 \sqrt{x} + k. \]