If `(x)` is of the form `f(x)=a+b x+c x^2,` show that `int_0^1f(x)dx=1/6{f(0)+4f(1/2)+f(1)}`

If (x) is of the form f(x)=a+bx+cx^(2) show that int_(1)f(x)dx=(1)/(6){f(0)+4f((1)/(2))+f(1)}

If f(x)=x^(3) show that int_(a)^(b)f(x)dx=(b-a)/(6){f(a)+4f((a+b)/(2))+f(b)}

If f(x) is a quadratic polynomial in x such that 6int_0^1 f(x)dx-{f(0)+4f((1)/(2))}=kf(1) , then k=

(A) For each of the following function f(x), verify that: int_0^2 f(x) dx=int_0^1 f(x) dx+int_1^2 f(x) dx: (i) f(x)=x+2 (ii) f(x)= x^2+2 (iii) f(x)= e^x (B) For each of the following pairs of function f(x) and g(x) , verify that: int_0^1 [f(x)+g(x)]dx=int_0^1 f(x) dx+int_0^1 g(x) dx: (i) f(x)=1,g(x) = x^2 (ii) f(x)=e^x, g(x)=1

f(x) = min{2 sinx, 1- cos x, 1} then int_0^(pi)f(x) dx is equal to

f(0)=1 , f(2)=e^2 , f'(x)=f'(2-x) , then find the value of int_0^(2)f(x)dx

Let the function f be defined by f (x) = |x-1| -1/2, 0 le x le 2, f (x +2 ) = f (x) for all x in R. Evaluate (i) int _(0) ^(100) f (x) dx (ii) int _(0) ^(1)| f(2x ) |dx

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

If f(x)+f(3-x)=0 ,then int_(0)^(3)1/(1+2^(f(x)))dx=