Integrate the function f(x) = ax^2 + bx + c over the interval [0,1], where a, b and c are constants.

Firstly remember that d/dx(x^n) = nx^(n-1). And so the antiderivative, or integral of x^n, i.e. \int(x^n) = x^(n+1)/(n+1) + C (where C is the integration constant). When integrating with limits, i.e. when we define an interval that we're integrating over, we do not have to worry about the constant C, and so for example: \int(x^3) over [0,1] will be x^4/4 (x=1 - x=0), i.e. = 1^4/4 - 0^4/4 = 1/4.

Hence, for our given function f(x), \int(f(x)) over [0,1] will be ax^3/3 + bx^2/2 + cx/1 (x=1 - x=0) = a/3 + b/2 + c.