Prove that `underset(i=1)overset(n)Sigma(x_i-barx)=0`, where `barx` is the mean of `x_1,x_2,x_3,….x_n`.

Prove that following: underset(-1)overset(1)int(x-[x])dx

Prove that following: underset(0)overset(1)int|x-5|dx

Prove that following: underset(0)overset(4)int|x-1| dx

Prove that following: underset(-3)overset(1)int|x+2|dx

Evaluate: underset(-1)overset(2)int|x^3-x|dx

underset1overset3int(dx)/(x^2(x+1))

Prove that : underset(0)overset(2a)intf(x) dx = underset(0)overset(2a)intf(2a-x)dx

underset0overset1 int (1-x)/(1+x)dx

Prove that following: underset(1)overset(4)intf(x)dx, where f(x) ={ {:(4x+3, if 1 le x le2), (3x+5, if 2 le x le4)

Prove that following: underset(1)overset(3)intf(x) dx when f(x) = |x-1|+|x-2|+|x-3|