If `int_a^b(f(x)-3x)dx=a^2-b^2` then the value of `f(pi/6)` is ___

int_(a)^(b) f(x) dx =

Let A=[[3x^2], [1], [6x]],B=[a,b,c] and C=[[(x+2)^2, 5x^2, 2x],[5x^2, 2x,(x+2)^2],[ 2x,(x+2)^2, 5x^2]] be three given matrices, where a ,b ,ca n dx in Rdot Given that tr(AB)=tr(C). If f(x)=a x^2+b x+c , then the value of f(1) is ______.

If f(x) =(e^x)/(1+e^x), I_1=int(f(-a))^(f(a)) xg(x(1-x)dx, and I_2=int_(f(-a))^(f(a)) g(x(1-x))dx, then the value of (I_2)/(I_1) is (a) -1 (b) -2 (c) 2 (d) 1

Let T >0 be a fixed real number. Suppose f is continuous function such that for all x in R ,f(x+T)=f(x)dot If I=int_0^Tf(x)dx , then the value of int_3^(3+3T)f(2x)dx is (a) 3/2I (b) 2I (c) 3I (d) 6I

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 .

Evaluate: int_(-1)^4f(x)dx=4a n dint_2^4(3-f(x))dx=7, then find the value of int_2^(-1)f(x)dxdot

If f'(x)=f(x)+int_(0)^(1)f(x)dx ,given f(0)=1 , then the value of f(log_(e)2) is

If f^(prime)(x)=f(x)+int_0^1f(x)dx ,gi v e nf(0)=1, then the value of f((log)_2 2) is (a) 1/(3+e) (b) (5-e)/(3-e) (c) (2+e)/(e-2) (d) none of these

A continuous real function f satisfies f(2x)=3(f(x)AAx in RdotIfint_0^1f(x)dx=1, then find the value of int_1^2f(x)dx