If f is continuously differentiable function then
∫02.5 x2f′(x)dx is equal to
6f(6.25)+∑i∈{1,2,3,2,5} f(i)
6f(6.25)−∑i∈{1,2,3,2,5,6}
6f(2.5)−∑i∈{1,2,3,2,5,6}
6f(2.5)−∑i∈{1,2,3,5}
∫02.5 x2f′(x)dx
=∫01 x2f′(x)dx+∫12 x2f′(x)+∫23 x2f′(x)dx+∫32 x2f′(x)dx+∫25 x2f′(x)dx+∫56 x2f′(x)dx+∫625 x2f′(x)dx=0+∫12 f′(x)dx+2∫23 f′(x)dx+3∫32 f′(x)dx+4∫25 f′(x)dx+5∫56 f′(x)dx+6∫62.5 f′(x)dx
=(f(2)−f(1))+2(f(3)−f(2))+3(f(2)−f(3))+4(f(5)−f(2))+5[f(6)−f(5)]+6[f(2.5)−f(6)]=6f(2.5)−(f(1)+f(2)+f(3)+f(2)+(f(5)+f(6))