The value of $\int_{0}^{1} ([2 x] - 3) ([3 x] - 7) d x$ is,
where $[.]$ denotes the greatest integer function.

10

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\frac{46}{3}

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\frac{41}{3}

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\frac{37}{3}

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Solution

The correct option is B $\frac{46}{3}$
$\int_{0}^{1} ([2 x] - 3) ([3 x] - 7) d x = \int_{0}^{\frac{1}{3}} (- 3) (- 7) d x + \int_{\frac{1}{3}}^{\frac{1}{2}} (- 3) (1 - 7) d x + \int_{\frac{1}{2}}^{\frac{2}{3}} (1 - 3) (1 - 7) d x + \int_{\frac{2}{3}}^{1} (1 - 3) (2 - 7) d x = 21 \times \frac{1}{3} + 18 \times \frac{1}{6} + 12 \times \frac{1}{6} + 10 \times \frac{1}{3} = \frac{46}{3}$

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The value of ∫10([2x]−3)([3x]−7)dx is, where [.] denotes the greatest integer function.

The correct option is B 463 ∫10([2x]−3)([3x]−7)dx=∫130(−3)(−7)dx+∫1213(−3)(1−7)dx+∫2312(1−3)(1−7)dx+∫123(1−3)(2−7)dx=21×13+18×16+12×16+10×13=463

The value of $\int_{0}^{1} ([2 x] - 3) ([3 x] - 7) d x$ is,
where $[.]$ denotes the greatest integer function.