Integrating by parts. ∫ f (x) g " (x) dx - ∫ f " (x) g (x) dx = f (x) g ' (x) - ∫ f ' (x) g ' (x) dx - f ' (x) g (x) + ∫ f ' (x) g ' (x) dx = f (x) g ' (x) - f ' (x) g (x) Hence ,
1
∫
0
f (x) g " (x) dx -
1
∫
0
f " (x) g (x) dx = f (1) g ' (1) - f ' (1) g (1) - f (0) g ' (0) + f ' (0) g (0) = f (1) g ' (1) - f ' (1) g (1)