If `f(x)` is a function satisfying `f(x+a)+f(x)=0` for all `x epsilonR` and positive constant `a` such that `int_(b)^(c+b)f(x)dx` is independent to `b`, then find the least positive value of `c`.

If f(x) is a function satisfying f(x+a)+f(x)=0 for all x in R and positive constant a such that int_b^(c+b)f(x)dx is independent of b , then find the least positive value of c

If f(x) is a function satisfying f(x+a)+f(x)=0 for all x in R and positive constant a such that int_b^(c+b)f(x)dx is independent of b , then find the least positive value of cdot

If f(x) is a function satisfying f(1/x)+x^2f(x)=0 for all nonzero x , then evaluate int_(sintheta)^(cos e ctheta)f(x)dx

If f(x) is a function satisfying f((1)/(x))+x^(2)f(x)=0 for all non zero x then, int_(sin theta)^("cosec" theta) f(x)dx equals

Let f(x) and phi(x) are two continuous function on R satisfying phi(x)=int_(a)^(x)f(t)dt, a!=0 and another continuous function g(x) satisfying g(x+alpha)+g(x)=0AA x epsilonR, alpha gt0 , and int_(b)^(2k)g(t)dt is independent of b Least positive value fo c if c,k,b are n A.P. is

Let alpha + beta = 1, 2 alpha^(2) + 2beta^(2) = 1 and f(x) be a continuous function such that f(2 + x) + f(x) = 2 for all x in [0, 2] and p = int_(0)^(4) f(x) dx - 4, q = (alpha)/(beta) . Then, find the least positive integral value of 'a' for which the equation ax^(2) - bx + c = 0 has both roots lying between p and q, where a, b, c in N .

Let f(x) and phi(x) are two continuous function on R satisfying phi(x)=int_(a)^(x)f(t)dt, a!=0 and another continuous function g(x) satisfying g(x+alpha)+g(x)=0AA x epsilonR, alpha gt0 , and int_(b)^(2k)g(t)dt is independent of b If f(x) is an even function, then

Let f(x) and phi(x) are two continuous function on R satisfying phi(x)=int_(a)^(x)f(t)dt, a!=0 and another continuous function g(x) satisfying g(x+alpha)+g(x)=0AA x epsilonR, alpha gt0 , and int_(b)^(2k)g(t)dt is independent of b If f(x) is an odd function, then

If f(a+b-x) =f(x) , then int_(a)^(b) x f(x) dx is equal to

Let f(x) be a continuous and periodic function such that f(x)=f(x+T) for all xepsilonR,Tgt0 .If int_(-2T)^(a+5T)f(x)dx=19(ag0) and int_(0)^(T)f(x)dx=2 , then find the value of int_(0)^(a)f(x)dx .