Q. $\int_{}^{}{\frac{1 + x + \sqrt{x + x^{2}}}{\sqrt{x} + \sqrt{1 + x}}dx}$ is equal to
A) $\frac{1}{2}\ \sqrt{1 + x} + c$
B) $\frac{2}{3}(1 + x)^{\frac{3}{2}} + c$
C) $\sqrt{1 + x} + c$
D) $2(1 + x)^{\frac{3}{2}} + c$
Explanation
(b)
$\int_{}^{}{\frac{1 + x + \sqrt{x + x^{2}}}{\sqrt{x} + \sqrt{1 + x}}dx} = \int_{}^{}{\frac{\sqrt{(1 + x)^{2}} + \sqrt{x}\sqrt{x + 1}}{\sqrt{x} + \sqrt{1 + x}}\ dx}$
$= \int_{}^{}\sqrt{1 + x}\ dx = \frac{2}{3}(1 + x)^{\frac{3}{2}} + c$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.