Assume F, G : R → R are differentiable, and assume that F ′ and G′ are both continuous.
Consider two numbers x < y.
1. (8pts) Explain why F′G and FG′ are Riemann integrable on [x,y].
2. (10pts) Show that (you can use standard differentiation rules like the product rule, even if we did not reprove them in this course):
y y
F ′(s)G(s)ds = F (y)G(y) − F (x)G(x) − F (s)G′(s).
xx
Exercise 2 ( Integration by parts formula ( 18 pts ) ) . Assume F, G : RR -> RR are differentiable,and assume that !” and ” are both continuous . Consider two numbers ac < y .1 . ( 8 pts ) Explain why !”G and FG” are Riemann integrable on 2, 4).
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