The equation of the curve is `y=f(x)dot` The tangents at `[1,f(1),[2,f(2)],a n d[3,f(3)]` make angles `pi/6,pi/3,a n dpi/4,` respectively, with the positive direction of x-axis. Then the value of `int_2^3f^(prime)(x)f^(x)dx+int_1^3f^(x)dx` is equal to `-1/(sqrt(3))` (b) `1/(sqrt(3))` (e) 0 (d) none of these

The equation of the curve is y=f(x)dot The tangents at [1,f(1),[2,f(2)],a n d[3,f(3)] make angles pi/6,pi/3,a n dpi/4, respectively, with the positive direction of x-axis. Then the value of int_2^3f^(prime)(x)f^(x)dx+int_1^3f^(x)dx is equal to (a)-1/(sqrt(3)) (b) 1/(sqrt(3)) (c) 0 (d) none of these

If f(x)=x^(3)+3x+4, then the value of int_(-1)^(1)f(x)dx+int_(0)^(4)f^(-1)(x)dx equals

Let y=f(x)=4x^(3)+2x-6 , then the value of int_(0)^(2)f(x)dx+int_(0)^(30)f^(-1)(y)dy is equal to _________.

If 2f(x) - 3 f(1//x) = x," then " int_(1)^(2) f(x) dx is equal to

if int_(0)^(1)f(x)dx=1 and f(2x)=2f(x) then 3int_(1)^(2)f(x)dx is equal to

The tangent to the graph of the function y=f(x) at the point with abscissae x=1, x=2, x=3 make angles pi/6,pi/3 and pi/4 respectively. The value of int_1^3f\'(x)f\'\'(x)dx+int_2^3f\'\'(x)dx is (A) (4-3sqrt(3))/3 (B) (4sqrt(3)-1)/(3sqrt(3)) (C) (4-3sqrt(3))/2 (D) (3sqrt(3)-1)/2

A continous function f(x) is such that f(3x)=2f(x), AA x in R . If int_(0)^(1)f(x)dx=1, then int_(1)^(3)f(x)dx is equal to

Let y=f(x) be a thrice differentiable function in (-5,5) .Let the tangents to the curve y=f(x) at (1,f(1)) and (3,f(3)) make angles pi/6 and pi/4 ,respectively with positive "x" -axis.If 27int_(1)^(3)((f'(t))^(2)+1)f''(t)dt=alpha+beta sqrt(3) where alpha,beta are integers,then the value of alpha+beta equals