_0^1 dx 1 + x - x = - Vidyadip

MCQ

$\int_0^1 {\frac{{dx}}{{\sqrt {1 + x} - \sqrt x }} = } $

A
$\frac{{2\sqrt 2 }}{3}$
✓
$\frac{{4\sqrt 2 }}{3}$
C
$\frac{{8\sqrt 2 }}{3}$
D
None of these

✓

Answer

Correct option: B.

$\frac{{4\sqrt 2 }}{3}$

b
(b) $I = \int_0^1 {\frac{{dx}}{{\sqrt {1 + x} - \sqrt x }} = \int_0^1 {\frac{{(\sqrt {1 + x} + \sqrt x )dx}}{{(\sqrt {1 + x} - \sqrt x )(\sqrt {1 + x} + \sqrt x )}}} } $

$ = \int_0^1 {\frac{{(\sqrt {1 + x} + \sqrt x )}}{{1 + x - x}}} dx = \int_0^1 {\sqrt {1 + x\,} dx} + \int_0^1 {\sqrt x } dx $

$= \frac{{4\sqrt 2 }}{3}$.

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