Let `f` be a non-negative function defined on the interval `[0,1]`. If `int_0^xsqrt(1-(f^(prime)(t))^2)dt=int_0^xf(t)dt ,0lt=xlt=1,a n d \ f(0)=0`, then

Let f be a real-valued function defined on the inverval (-1,1) such that e^(-x)f(x)=2+int_0^xsqrt(t^4+1)dt , for all, x in (-1,1) and let f^(-1) be the inverse function of fdot Then (f^(-1))^'(2) is equal to (a) 1 (b) 1/3 (c) 1/2 (d) 1/e

If f(x)=int_0^x(sint)/t dt ,x >0, then

Let f be a real-valued function defined on interval (0,oo) ,by f(x)=lnx+int_0^xsqrt(1+sint).dt . Then which of the following statement(s) is (are) true? (A). f"(x) exists for all in (0,oo) . " " (B). f'(x) exists for all x in (0,oo) and f' is continuous on (0,oo) , but not differentiable on (0,oo) . " " (C). there exists alpha>1 such that |f'(x)|<|f(x)| for all x in (alpha,oo) . " " (D). there exists beta>1 such that |f(x)|+|f'(x)|<=beta for all x in (0,oo) .

If int_0^xf(t) dt=x+int_x^1 tf(t)dt, then the value of f(1)

Ifint_(pi/3)^xsqrt((3-sin^2t))dt+int_0^ycostdt=0,then evaluate (dy)/(dx)

If m, n in R , then the value of I(m,n)=int_(0)^(1) t^(m)(1+t)^(n)dt is -

Let f(x) be a non-constant twice differentiable function defined on (-oo,oo) such that f(x)=f(1-x)a n df^(prime)(1/4)=0. Then (a) f^(prime)(x) vanishes at least twice on [0,1] (b) f^(prime)(1/2)=0 (c) int_(-1/2)^(1/2)f(x+1/2)sinxdx=0 (d) int_(-1/2)^(1/2)f(t)e^(sinpit)dt=int_(1/2)^1f(1-t)e^(sinpit)dt

Let f:(0,oo)vec(0,oo) be a differentiable function satisfying, xint_0^x(1-t)f(t)dt=int_0^x tf(t)dtx in R^+a n df(1)=1. Determine f(x)dot

Let f be a differentiable function satisfying int_(0)^(f(x))f^(-1)(t)dt-int_(0)^(x)(cost-f(t)dt=0 and f((pi)/2)=2/(pi) The value of int_(0)^(pi//2) f(x)dx lies in the interval

The value of int_0^(sin^2x)sin^(-1)(sqrtt)dt+int_0^(cos^2x)cos^(-1)(sqrtt)dt is