# Chapter 7: Chapter 7

Problem 195

Find \(\mathrm{F}^{\prime}(\mathrm{x})\) where a) \(F(x)=x \int_{0} e^{-(x) 2(t) 2} d t\) b) \(F(x)=\sin x \int_{(x) 2}\left(x^{2}-t^{2}\right)^{n} d t\)

Problem 197

Proceed from the defintion of the Stieltjes integral to show that the function f given by $$ \begin{array}{cc} \mathrm{f}(\mathrm{x})=\mathrm{g}(\mathrm{x})=0 & 0 \leq \mathrm{x} \leq 1 \\ \text { and }=1 & 1<\mathrm{x} \leq 2 \end{array} $$ is not Stieltjes integrable with respect to \(\mathrm{g}\).

Problem 198

Evaluate the Stieltjes integral $^{1} \int_{-1} \mathrm{x} \mathrm{d}|\mathrm{x}|$.

Problem 199

Let \(\mathrm{f}\) be a function from \([\mathrm{a}, \mathrm{b}]\) into \(\mathrm{R}\) which is continuous at \(c \in[a, b]\) and let \(X_{c}\) be the characteristic function of \(c\), i.e., $$ \begin{array}{cc} X_{c}(x)=1 & x=c \\ \text { and }=0 & x \neq c . \end{array} $$ Show, using the definition of the Stieltjes integral that $b_{a} f d X_{c}=0 \quad c \in(a, b)$ $$ \begin{array}{ll} \text { and }=-\mathrm{f}(\mathrm{a}) & \mathrm{c}=\mathrm{a} \\ \text { and }=\mathrm{f}(\mathrm{b}) & \mathrm{c}=\mathrm{b} \text { . } \end{array} $$

Problem 201

Suppose \(g\) is a continuously differentiable monotonically increasing function on \([a, b]\) and \(f\) is bounded on \([a, b]\). Prove that $$ b \int_{a} f d g=b \int_{a} f(x) g^{\prime}(x) d x $$ Use this result to find the total mass of a linear distribution on \([a, b]\) with a continuous density function \(\rho(x)\).