Let [t] denote the greatest integer less than or equal to t. Then the value of ∫ limits12|2 x-[3 x]| d x is

Question

Let [t] denote the greatest integer less than or equal to t. Then the value of ∫ limits12|2 x-[3 x]| d x is (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

3<3 x<6 Take cases when 3<3 x<4,4<3 x<5 5<3 x<6 Now ∫ limits12|2 x-[3 x]| d x =∫ limits14 / 3(3-2 x) d x+∫ limits4 / 35 / 3(4-2 x) d x+∫ limits5 / 32(5-2 x) d x =(2/9)+(3/9)+(4/9)=1

Q. Let [t] denote the greatest integer less than or equal to $t$ . Then the value of $1 \int 2 ∣2 x - [3 x] ∣ d x$ is___

$3 < 3 x < 6$
Take cases when $3 < 3 x < 4, 4 < 3 x < 5$
$5 < 3 x < 6$
Now $1 \int 2 ∣2 x - [3 x] ∣ d x$
$= 1 \int 4/3 (3 - 2 x) d x + 4/3 \int 5/3 (4 - 2 x) d x + 5/3 \int 2 (5 - 2 x) d x$
$= \frac{2}{9} + \frac{3}{9} + \frac{4}{9} = 1$

Q. Let [t] denote the greatest integer less than or equal to t. Then the value of 1∫2​∣2x−[3x]∣dx is___

3<3x<6 Take cases when 3<3x<4,4<3x<5 5<3x<6 Now 1∫2​∣2x−[3x]∣dx =1∫4/3​(3−2x)dx+4/3∫5/3​(4−2x)dx+5/3∫2​(5−2x)dx =92​+93​+94​=1

Q. Let [t] denote the greatest integer less than or equal to $t$ . Then the value of $1 \int 2 ∣2 x - [3 x] ∣ d x$ is___

$3 < 3 x < 6$
Take cases when $3 < 3 x < 4, 4 < 3 x < 5$
$5 < 3 x < 6$
Now $1 \int 2 ∣2 x - [3 x] ∣ d x$
$= 1 \int 4/3 (3 - 2 x) d x + 4/3 \int 5/3 (4 - 2 x) d x + 5/3 \int 2 (5 - 2 x) d x$
$= \frac{2}{9} + \frac{3}{9} + \frac{4}{9} = 1$