int_(0)^( alpha/3)(f(x))/(f(x)+f((alpha-3x)/(3)))dx=

Let int_(alpha)^( beta)(f(alpha+beta-x))/(f(x)+f(alpha+beta-x))dx=4 then

f(x)>0AAx in R and is bounded. If lim_(n->oo)[int_0^a(f(x)dx)/(f(x)+f(a-x))+aint_a^(2a)(f(x)dx)/(f(x)+f(3a-x)) +a^2int_(2a)^(3a)(f(x)dx)/(f(x)+f(5a-x))+...+a^(n-1)int_((n-1)a)^(n a)(f(x)dx)/(f(x)+f[(2n-1)a-x]]] =7//5 (where a<1), then a is equal to

f(x)>0AAx in R and is bounded. If lim_(n->oo)[int_0^a(f(x)dx)/(f(x)+f(a-x))+aint_a^(2a)(f(x)dx)/(f(x)+f(3a-x)) +a^2int_(2a)^(3a)(f(x)dx)/(f(x)+f(5a-x))+...+a^(n-1)int_((n-1)a)^(n a)(f(x)dx)/(f(x)+f[(2n-1)a-x]]] =7//5 (where a<1), then a is equal to

P(alpha,f(alpha)) and Q(beta,f(beta)) are ends of an arc in the first quadrant.The area bounded by the arc,ordinates through P and Q, and the x -axis is (A) int_(f(alpha))^(f(beta))f^(-1)(y)dy(B)int_(alpha)^( beta)f^(-1)(y)dy(C)int_(alpha)^( beta)f(x)dx(D)int_(f(alpha))^(f(beta))f(x)dx

If int ((x cos alpha+1)dx)/((x^(2)+2x cos alpha+1)^(3//2))=(f(x))/(sqrt(x^(2)+2x cos alpha+ 1))+c then f(x)=

If int(x cos alpha+1)/((x^(2)+2x cos alpha+1)^((3)/(2)))dx=(f(x))/(sqrt(g(x)))+c then find f(x)

Match the following {:("List - 1","List - II"),("I) "int_(-1)^(1)x|x|dx,"a) "(pi)/(2)),("II) "int_(0)^((pi)/(2))(1+log((4+3sinx)/(4+3cosx)))dx,"b) "int_(0)^((pi)/(2))f(x)dx),("III) "int_(0)^(a)f(x)dx,"c) "int_(0)^(a)[f(x)+f(-x)]dx),("IV) "int_(-a)^(a)f(x)dx,"d) "0),(,"e) "int_(0)^(a)f(a-x)dx):}

int_(2)^(3)f(5-x)dx-int_(2)^(3)f(x)dx=

Let a function f(x,alpha) be continuous for a<=x<=b and 0<=a<=d. Then,for any alpha in[c,d], if I(alpha)=int_(a)^(b)f(x,alpha)dx, then (d)/(d alpha)(I(alpha))=int_(a)^(b)rho(f(x,alpha))/(rho alpha)dx

Let g(x)=|(f(x+alpha), f(x+2a), f(x+3alpha)), f(alpha), f(2alpha), f(3alpha),(f\'(alpha),(f\'(2alpha), f\'(3alpha))| , where alpha is a constant then Lt_(xrarr0(g(x))/x= (A) 0 (B) 1 (C) -1 (D) none of these