`int_(0)^(1)(x+x^(2))dx+int_(1)^(2)(x^(2)+x^(3))dx`

int_(0)^(3)(x^(2)+1)dx

int_(0)^(1)x^2dx

int_(1)^(2) (x^3+1) dx

1. int_(0)^(2)3x^(2)dx

Let f:[0,1]to R be a continuous function then the maximum value of int_(0)^(1)f(x).x^(2)dx-int_(0)^(1)x.(f(x))^(2)dx for all such function(s) is:

int_(0)^(2) (1)/(4+x+x^(2))dx

If I_(1)=int_(0)^(1) 2^(x^(2)) dx, I_(2)=int_(0)^(1) 2^(x^(3)) dx, I_(3)=int_(1)^(2) 2^(x^(2))dx and I_(4)=int_(1)^(2) 2^(x^(3))dx then

int_(0)^(1) (1)/(1+x+2x^(2))dx

int_(0)^(2)(x+2)/(x+1)dx is

If int_(0)^(1) x e^(x^(2) ) dx=alpha int_(0)^(1) e^(x^(2)) dx , then