Prove that `int_0^x[t]dt=([x]([x]-1))/2+[x](x-[x]),` where [.] denotes the greatest integer function.

Prove that int_0^xf[t]dt=([x]([x]-1))/2+[x](x-[x]), where [.] denotes the greatest integer function.

(i) The set of all values of a for which f(x)=int_(0)^(x){-2(a^(2)-3a+2)sin2x+(a-1)}dx doesn't posses any critical point. (ii) Prove that int_(0)^(x)[t.]dt=([x]([x]-1))/(2)+[x](x-[x]) , where [x] denotes greatest integer function.

Prove that int[t]dt=([x][x]-1))/(2)+[x](x-[x]) where [.] denotes the greatest integer function.

Prove that int_(0)^(x) [x]dx=[x]([x]-1)/(2)+[x] (x-[x]) , where [.] denotes the greatest integer function

lim_(xrarr1([x]+[x]) , (where [.] denotes the greatest integer function )

int_(-1)^(41//2)e^(2x-[2x])dx , where [*] denotes the greatest integer function.

Let f(x)=(-1)^([x]) where [.] denotes the greatest integer function),then

If [log_(2)((x)/([x]))]>=0, where [.] denote the greatest integer function,then

Lt_(x to oo) ([x])/(x) (where[.] denotes greatest integer function )=