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7.2 definite integral — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths7.2 definite integral1 Mark
MCQ
$\mathop {Limit}\limits_{x\,\, \to \,\,{x_1}} \,\,\frac{x}{{x\,\, - \,\,{x_1}}}\,\,\,\int\limits_{{x_1}}^x {\,f(t)} \, dt$ is equal to :
  • A
    $\frac{{f\,\,\left( {{x_1}} \right)}}{{{x_1}}}$
  • ✓
    $x_1 f (x_1)$
  • C
    $f (x_1)$
  • D
    does not exist
✓

Answer

Correct option: B.
$x_1 f (x_1)$
b
$\mathop {Limit}\limits_{x \to {x_1}} \,\,\,\frac{{\int\limits_{{x_1}}^x {f(t)} \,\,dt}}{{\left( {\frac{{x - {x_1}}}{x}} \right)}}$ $= \mathop {Limit}\limits_{x \to {x_1}} \,\,\,\frac{{f(x)\,.\,{x^2}}}{{{x_1}}}\,$   (using Lopital’s rule)

$= x_1\, f (x_1) \Rightarrow B$

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