`int_0^(102)(x-1)(x-2)..(x-100)xx(1/(x-1)+1/(x-2)+..1/(x-100))dx=101 !-100 !`

prove that int_0^(102)(x-1)(x-2)..(x-100)xx1/(1/(x-1)+1/(x-2)+. .1/(x-100))dx=101 !-100 !

Prove that int_0^(102)(x-1)(x-2)(x-100) x(1/((x-1)+1/((x-2))+1/((x-100))dx=101 !-100 !

int_(0)^(102)(x-1)(x-2)dots(x-100)xx((1)/(x-1)+(1)/(x-2)+.(.1)/(x-100))dx=101!-100

int_0^1x(1-x)^100dx

Let f(x)=(a^(x))/(a^(x)+sqrt(a)),a>1 and int_(0)^(2)(f((1)/(100))+f((2)/(100))+f((3)/(100))+...+f((99)/(100)))*dx=k times11 then k=

The vlaue of ((5050)int_(0)^(1)(1-x^(50))^(100)dx)/(int_(0)^(1) (1-x^(50))^(100)dx) , is

int(1)/(x(x^(100)+1))*dx=....+c;(x>0)

If f(x)=|(1,x,x+1),(2x,x(x-1),(x+1)x),(3x(x-1),x(x-1)(x-2),(x+1)x(x-1))|, then f(100) is equal to - (i)0 (ii)1 (iii)100 (iv)-100