_ ^ dx x + a + x + b = - Vidyadip

MCQ

$\int_{}^{} {\frac{{dx}}{{\sqrt {x + a} + \sqrt {x + b} }}} = $

A
$\frac{2}{{3(b - a)}}[{(x + a)^{3/2}} - {(x + b)^{3/2}}] + c$
✓
$\frac{2}{{3(a - b)}}[{(x + a)^{3/2}} - {(x + b)^{3/2}}] + c$
C
$\frac{2}{{3(a - b)}}[{(x + a)^{3/2}} + {(x + b)^{3/2}}] + c$
D
None of these

✓

Answer

Correct option: B.

$\frac{2}{{3(a - b)}}[{(x + a)^{3/2}} - {(x + b)^{3/2}}] + c$

b
(b)$\int_{}^{} {\frac{{dx}}{{\sqrt {x + a} + \sqrt {x + b} }} = \int_{}^{} {\frac{{\sqrt {x + a} - \sqrt {x + b} }}{{(x + a) - (x + b)}}\,dx} } $

$ = \frac{1}{{(a - b)}}\int_{}^{} {{{(x + a)}^{1/2}}dx} - \frac{1}{{(a - b)}}\int_{}^{} {{{(x + b)}^{1/2}}dx} $

$[ = \frac{2}{{3(a - b)}}[{(x + a)^{3/2}} - {(x + b)^{3/2}}] + c.]$

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