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7.2 definite integral — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths7.2 definite integral1 Mark
MCQ
$\int_{0}^{2}\left(\left|2 x^{2}-3 x\right|+\left[x-\frac{1}{2}\right]\right) d x$,where  is the greatest integer function, is equal to.
  • A
    $\frac{7}{6}$
  • ✓
    $\frac{19}{12}$
  • C
    $\frac{31}{12}$
  • D
    $\frac{3}{2}$
✓

Answer

Correct option: B.
$\frac{19}{12}$
b
$\int_{0}^{2}\left|2 x ^{2}-3 x \right| dx$

$=\int_{0}^{\frac{3}{2}}\left(3 x -2 x ^{2}\right) dx +\int_{\frac{3}{2}}^{2}\left(2 x ^{2}-3 x \right) dx =\frac{19}{12} .$

$\int_{0}^{2}\left[ x -\frac{1}{2}\right] dx =\int_{\frac{-1}{2}}^{\frac{3}{2}}[ t ] dt$

$=\int_{-\frac{1}{2}}^{0}(-1) dt +\int_{0}^{1} 0 \cdot dt +\int_{1}^{\frac{3}{2}} 1 \cdot dt =0 .$

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