Let `F(x)=int_x^[x^2+pi/6][2cos^2t.dt]` for all `x in R` and `f:[0,1/2] -> [0,oo)` be a continuous function.For `a in [0,1/2]`, if F'(a)+2 is the area of the region bounded by x=0,y=0,y=f(x) and x=a, then f(0) is

Let f:[-1,2]->[0,oo) be a continuous function such that f(x)=f(1-x)fora l lx in [-1,2]dot Let R_1=int_(-1)^2xf(x)dx , and R_2 be the area of the region bounded by y=f(x),x=-1,x=2, and the x-axis . Then (a) R_1=2R_2 (b) R_1=3R_2 (c) 2R_1 (d) 3R_1=R_2

If a continuous function f on [0, a] satisfies f(x)f(a-x)=1, a >0, then find the value of int_0^a(dx)/(1+f(x))

A continuous function f(x) satisfies the relation f(x)=e^x+int_0^1 e^xf(t)dt then f(1)=

Let f(x)= maximum {x^2, (1-x)^2, 2x(1 - x)} where x in [0, 1]. Determine the area of the region bounded by the curve y=f(x) and the lines y = 0,x=0, x=1.

Let a function f be defined by f(x)=(x-|x|)/x for x ne 0 and f(0)=2. Then f is

Let f(x) =M a xi mu m {x^2,(1-x)^2,2x(1-x)}, where 0lt=xlt=1. Determine the area of the region bounded by the curves y=f(x) ,x-axis ,x=0, and x=1.

Let f:R -(0,oo) be a real valued function satisfying int_0^x tf(x-t) dt =e^(2x)-1 then f(x) is

If" f, is a continuous function with int_0^x f(t) dt->oo as |x|->oo then show that every line y = mx intersects the curve y^2 + int_0^x f(t) dt = 2

Let f(x)=1/x^2 int_0^x (4t^2-2f'(t))dt then find f'(4)

Let a function f be defined by f (x) =[x-|x|]/x for xne 0 and f(0)=2 .Then f is :