Question

J I [lx ll+lx 21+1x 31]dr

Answer 1

The given integral is,

I= ∫ 1 4 [ | x−1 |+| x−2 |+| x−3 | ]dx

Let, I= I 1 + I 2 + I 3 (1)

Where,

I 1 = ∫ 1 4 | x−1 |dx I 2 = ∫ 1 4 | x−2 |dx I 3 = ∫ 1 4 | x−3 |dx

Consider I 1 = ∫ 1 4 | x−1 |dx , ( x−1 )≥0for 1≤x≤4.

I 1 = ∫ 1 4 ( x−1 )dx = [ x 2 2 −x ] 1 4 =[ 8−4− 1 2 +1 ] = 9 2

Consider I 2 = ∫ 1 4 | x−2 |dx , ( x−2 )≥0 for 2≤x≤4 and ( x−2 )≤0for 1≤x≤2.

I 2 = ∫ 1 2 ( 2−x )dx + ∫ 2 4 ( x−2 )dx = [ 2x− x 2 2 ] 1 2 + [ x 2 2 −2x ] 1 2 =[ 4−2−2+ 1 2 ]+[ 8−8−2+4 ] = 5 2

Consider I 3 = ∫ 1 4 | x−3 |dx , ( x−3 )≥0 for 3≤x≤4 and ( x−3 )≤0 for 1≤x≤3.

I 3 =− ∫ 1 3 ( x−3 )dx + ∫ 3 4 ( x−3 )dx = [ 3x− x 2 2 ] 1 3 + [ x 2 2 −3x ] 3 4 =[ 9− 9 2 −3+ 1 2 ]+[ 8−12− 9 2 +9 ] = 5 2

Substitute the values of I 1 , I 2 and I 3 in equation (1),

I= 9 2 + 5 2 + 5 2 = 19 2

Thus, the value of I is 19 2 .