Determine if the following improper integrals of the first kind converge or diverge: a) \({ }^{\infty} \int_{1}\left(1 / \mathrm{x}^{\mathrm{p}}\right) \mathrm{dx}\) b) \({ }^{\infty} \int_{0} \mathrm{e}^{-\mathrm{rx}} \mathrm{dx}\) c) \(^{\infty} \int_{0} \sin x d x\).

a) Given \(\mathrm{r}=\left[\left(\mathrm{x}-\mathrm{x}_{0}\right)^{2}+\left(\mathrm{y}-\mathrm{y}_{0}\right)^{2}\right]\), show that the integral \(\iint_{D}\left(1 / r^{m}\right) d A\), where \(\left(x_{0}, y_{0}\right)\) is a Point of \(D\) and \(m>0\) (so that the integral is improper), is convergent if \(\mathrm{m}<2\) (D is a circle with center at $\left(\mathrm{x}_{0}, \mathrm{y}_{0}\right)\( and with radius \)\mathrm{c}$ ). b) Show that \(\iint_{D} \sin (\mathrm{y} / \mathrm{x}) \mathrm{d} \mathrm{A}\) is absolutely convergent, where D is the square domain \(0<\mathrm{x}<1,0<\mathrm{y}<1\)

Evaluate, for any constant \(c>0\) $$ \infty \int_{-\infty} e^{-c(x) 2} d x $$ and using this, evaluate $$ \int_{(R) n} e^{-\langle x, x} d x_{1} d x_{2} \ldots d x_{n} $$ where \(x \in R^{n}, P\) is a positive definite symmetric matrix, and \(<>\) denotes the Euclidean inner product (i.e., \(<\mathrm{Tx}, \mathrm{x}>\) is a positive definite quadratic form).

Test the following integrals for convergence: a) \(^{\infty} \int_{0}[\mathrm{dx} /\\{(1+\mathrm{x}) \sqrt{\mathrm{x}}\\}]\) b) \(^{\infty} \int_{-\infty}[\mathrm{dx} /\\{\mathrm{x}(\mathrm{x}-1)\\}]\)

Test the following integrals of the first kind for convergence: a) \({ }^{\infty} \int_{0}(1 / \mathrm{t}) \sin \mathrm{t} \mathrm{dt}\) b) \(^{\infty} \int_{0} \sin u^{2}\) du .

Prove the following theorems (assume \(\mathrm{f}(\mathrm{x}) \in \mathrm{C}\) ): a) For \(\mathrm{a} \leq \mathrm{x}<\infty\), if $^{\infty} \int_{\mathrm{a}}|\mathrm{f}(\mathrm{x})| \mathrm{dx}\( converges then \){ }^{\infty} \int_{\mathrm{a}} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}$ converges. b) For \(\mathrm{a} \leq \mathrm{x}<\infty\), if $\lim _{\mathrm{x} \rightarrow \infty} \mathrm{x}^{\mathrm{p}} \mathrm{f}(\mathrm{x})$ exists for \(\mathrm{p}>1\), then \(\infty_{a}|f(x)| d x\) converges. It is assumed that \(f\) is bounded on $[a, c]\( for every \)c>a$ c) For \(\mathrm{a}<\mathrm{x} \leq \mathrm{b}\), if $\lim _{\mathrm{x} \rightarrow \infty}^{+}(\mathrm{x}-\mathrm{a})^{\mathrm{p}} \mathrm{f}(\mathrm{x})\( exists for \)0<\mathrm{p}<1$, and f is bounded on \((a, b]\) then \(b_{a+}|f(x)| d x\) converges.