e^ x x , , ( x + x ) , ,dx , - Vidyadip

MCQ

$\int {\frac{{{e^{\sqrt x }}}}{{\sqrt x }}\,\,\left( {x + \sqrt x } \right)\,\,dx\,} $

✓
$2 {e^{\sqrt x }}\,\left[ {x\, - \,\sqrt x \, + 1\,} \right] + C$
B
${e^{\sqrt x }}\,\left[ {x\, - \,2\,\sqrt x \, + 1\,} \right]$
C
${e^{\sqrt x }}\,\left( {x + \sqrt x } \right)\,\, + \,\,C$
D
${e^{\sqrt x }}\,\left( {x + \sqrt x \, + \,1} \right)\,\, + \,\,C$

✓

Answer

Correct option: A.

$2 {e^{\sqrt x }}\,\left[ {x\, - \,\sqrt x \, + 1\,} \right] + C$

a
$\int {\frac{{{e^{\sqrt x }}}}{{\sqrt x }}\,\,\left( {x + \sqrt x } \right)\,\,dx\,} $ ;put $x = t^2$ ; $dx = 2t\, dt$
$= \int {{e^t}({t^2} + t)\,\,dt\,} $ $= e^t (At^2 + Bt + C)$ $(Let)$
Diffrentiate both the sides
$e^t (t^2 + t) = e^t (2At + B) + (At^2 + Bt + C) e^t$
On comparing coefficient we get $A = 1 ; B = - 1 ; C = 1 $

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