QJEEM21Ma0ccd: JEE Main 2021 Mathematics - Class 12 JEEM21Ma0ccd Solution: Let [t] denote the greatest integer t. Then the value of is…

Question

Verified Solution • 2025–26 Session moderate

Let [t] denote the greatest integer

\le

t. Then the value of

8.\int\limits_{ - {1 \over 2}}^1 {([2x] + |x|)dx}

is ___________.

Accepted Answer

I = \,\int\limits_{ - {1 \over 2}}^1 {([2x] + |x|)dx}

= \int\limits_{ - 1/2}^1 {[2x]\,dx + \int\limits_{ - 1/2}^1 {|x|\,} dx}

= 0 + \int\limits_{ - 1/2}^0 {( - x)\,} dx + \int\limits_0^1 {x\,} dx

= \left( { - {{{x^2}} \over 2}} \right)_{ - 1/2}^0 + \left( {{{{x^2}} \over 2}} \right)_0^1

= \left( {0 + {1 \over 8}} \right) + {1 \over 2}

= {5 \over 8}

\therefore

8I = 5

Step-by-Step Verified Solution