_ 0 ^ 2 ( |2 x^ 2 -3 x |+ [x-1 2 ] ) d x,where is the greatest in… - Vidyadip

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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