Exercise 7.5.6

[Integration-by-parts]

(a)

Assume $h(x)$ and $k(x)$ have continuous derivatives on $[a,b]$ and derive the familiar integration-by-parts formula $${\int \; <!--\; nolimits\; -->}_a^{b}h(t)k^{\prime }(t)\mathit{dt}=h(b)k(b)-h(a)k(a)-{\int \; <!--\; nolimits\; -->}_a^b{h}^{\prime }(t)k(t)\mathit{dt}$$

(b)

Explain how the result in Exercise 7.4.6 can be used to slightly weaken the hypothesis in part (a).