_0^a x ,dx a^2 + x^2 = - Vidyadip

MCQ

$\int_0^a {\frac{{x\,dx}}{{\sqrt {{a^2} + {x^2}} }}} = $

✓
$a\,(\sqrt 2 - 1)$
B
$a\,(1 - \sqrt 2 )$
C
$a\,(1 + \sqrt 2 )$
D
$2a\sqrt 3 $

✓

Answer

Correct option: A.

$a\,(\sqrt 2 - 1)$

a
(a) Put $t = {a^2} + {x^2} $

$\Rightarrow 2xdx = dt,$ then

$\int_0^a {\frac{{xdx}}{{\sqrt {{a^2} + {x^2}} }} = \frac{1}{2}\int_{{a^2}}^{2{a^2}} {\frac{1}{{\sqrt t }}dt} } $

$ = [{(2{a^2})^{1/2}} - {a^{2/2}}] = a(\sqrt 2 - 1)$.

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