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7.1 inderfinite integral — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths7.1 inderfinite integral1 Mark
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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