The equation `int_0^x (t^2-8t+13)dt=xsin(a/x)` has a solution if `sin(a/6)` is (A) zero (B) `-1` (C) `1` (D) none of these

Cosider the equation int_(0)^(x) (t^(2)-8t+13)dt= x sin (a//x) If x takes the values for which the equation has a solution, then the number of values of a in [0, 100] is

If [x] denotes the integral part of x and k=sin^-1 ((1+t^2)/(2t))gt0 then integral valueof alpha for which the equation (x-[k])(x+alpha)-1=0 has integral roots is (A) 1 (B) 2 (C) 4 (D) none of these

Cosider the equation int_(0)^(x) (t^(2)-8t+13)dt= x sin (a//x) The number of real values of x for which the equation has solution is

If f(x)=int_0^1(dt)/(1+|x-t|) ,then f^(prime)(1/2) is equal to (a)0 (b) 1/2 (c) 1 (d) none of these

If int_(0)^(x)f(t)dt=x+int_(x)^(1)f(t)dt ,then the value of f(1) is

The solution for x of the equation int_(sqrt(2))^(x)(dt)/(sqrt(t^(2)-1))=(pi)/(2) is pi(b)(sqrt(3))/(2) (c) 2sqrt(2) (d) none of these

int_0^1 (tan^-1x)/xdx-1/2int_0^(pi/2) t/sint dt has the value (A) -1 (B) 1 (C) 2 (D) 0

Let f(x)=int_0^x (sin^100t)/(sin^100t+cos^100t)dt then f(2pi)= (A) 2f(pi/2) (B) 4f(pi/2) (C) f(pi/2) (D) none of these