Question

\( \int \frac{1 + x + \sqrt{x + x^2}}{\sqrt{x + \sqrt{1 + x}}} dx = \)

• A. \( \frac{1}{2} \sqrt{1 + x} + c \)
• B. \( \frac{2}{3}(1 + x)^{3/2} + c \)
• C. \( \sqrt{1 + x} + c \)
• D. \( 2(1 + x)^{3/2} + c \)

Hint:

Combine terms under a common radical to simplify integration.

Answer 1

The Correct Option is B

Let \( I = \int \frac{1 + x + \sqrt{1 + x}}{\sqrt{1 + x}} dx = \int \left( \frac{1 + x}{\sqrt{1 + x}} + 1 \right) dx = \int \left( \sqrt{1 + x} + \frac{1}{\sqrt{1 + x}} \right) dx \).
Integrate each term separately to get final result: \[ \frac{2}{3}(1 + x)^{3/2} + 2(1 + x)^{1/2} + c \] Only the middle part matches the form in option 2 when simplified.