Chapter 10

Determine if the following integrals are proper or improper. If an integral is improper, determine if it is of the first, second or third kind. a) \(^{-1} \int_{-\infty}[\mathrm{dx} /\\{\mathrm{x}(\mathrm{x}-1)\\}]\) b) \(^{\infty} \int_{0}\left[\left(\mathrm{e}^{-\mathrm{at}}-\mathrm{e}^{-\mathrm{bt}}\right) / \mathrm{t}\right] \mathrm{dt}\) c) \(1 / 2 \int_{0}[\mathrm{dx} /\\{\mathrm{x}(\mathrm{x}-1)\\}]\) d) \(8 \int_{4}\left[(x d x) /\left\\{(x-3)^{2}\right\\}\right]\) e) \(^{\infty} \int_{0}\left(e^{-x} / \sqrt{x}\right) d x\).

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) \(^{\infty} \int_{0} \sin x d x\).

Test for convergence: a) \(^{\infty} \int_{2}\left[\left(x^{2} d x\right) / \sqrt{\left(x^{7}+1\right)}\right]\) b) \(^{\infty} \int_{2}\left[\left(x^{3} d x\right) / \sqrt{ \left.\left(x^{7}+1\right)\right]}\right.\) c) \(^{\infty} \int_{2}\left[\left(x^{2}+4 x+4\right) /\left\\{(\sqrt{(x-1)})^{3} \sqrt{\left. \left.\left(x^{3}-1\right)\right\\}\right]} \mathrm{dx}\right.\right.\).

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 .

Determine if the following improper integrals of the second kind are convergent: a) \(1 \int_{0}\left[\mathrm{dx} /\left\\{\left(1-\mathrm{x}^{3}\right)^{1 / 3}\right\\}\right]\) b) \(\pi \int_{0}\left[(\sin \mathrm{x}) / \mathrm{x}^{3}\right]\) c) \(^{1} \int_{0} \ln [1 /(1-x)] d x\)

Determine for what values of \(\mathrm{x}\) the integral $$ { }^{1} \int_{0} \mathrm{e}^{-t} \mathrm{t}^{\mathrm{x}-1} \mathrm{dt} $$ is a) proper; b) improper, but convergent.

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

Given \(\mathrm{n}\) is a real number, show that $$ \infty \int_{0}^{\infty} x^{n} e^{-x} d x $$ converges when \(\mathrm{n}>-1\) and diverges when \(\mathrm{n} \leq-1\).

a) Given that \(^{\infty} \int_{1}[\mathrm{f}(\mathrm{t})]^{2} \mathrm{t}^{-2}\) dt converges, prove \({ }^{\infty} \int_{1}|\mathrm{f}(\mathrm{t})| \mathrm{t}^{-2} \mathrm{dt}\) converges. b) Given that \(\mathrm{f}(\mathrm{t})\) has a singularity at \(\mathrm{t}=0\) and that \({ }^{1} \int_{0}^{+}|\mathrm{f}(\mathrm{t})|^{\mathrm{b}}\) dt converges, prove that \({ }^{1} \int_{0}^{+}|\mathrm{f}(\mathrm{t})|^{\mathrm{a}} \mathrm{dt}\) converges, where \(0<\mathrm{a}<\mathrm{b}\).

Determine if the following improper integrals converge absolutely: a) \(\left.\int_{1}\left[(\sin x) / x^{2}\right] d x b\right)^{\infty} \int_{1}[(\cos x) / x] d x\) c) \(^{\infty} \int_{0}[(\sin x) / x] d x\).

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.

Given that: (i) \(g(x) \in C\) (the set of continuous functions) for \(a \leq x<\infty\) (ii) \(\mathrm{g}(\mathrm{x})\) is a nonincreasing function for \(\mathrm{a} \leq \mathrm{x}<\infty\) (iii) \(\lim _{\mathrm{x} \rightarrow \infty} \mathrm{g}(\mathrm{x})=0\) Prove that: a) \(^{\infty} \int_{a} g(x) \sin x d x\) converges. b) \(^{\infty} \int_{\mathrm{a}} \mathrm{g}(\mathrm{x})|\sin \mathrm{x}| \mathrm{dx}\) converges if \(^{\infty} \int_{\mathrm{a}} \mathrm{g}(\mathrm{x}) \mathrm{dx}\) converges and it diverges if \(^{\infty} \int_{\mathrm{a}} \mathrm{g}(\mathrm{x}) \mathrm{d} \mathrm{x}\) diverges.

Determine if the following integrals converge absolutely: a) \(\int_{0}\left[(\cos x) / \sqrt{ \left.\left(1+x^{3}\right)\right]} d x\right.\) b) \(^{\infty} \int_{0} \sin x^{2} d x\).

Determine if the following integrals converge absolutely, conditionally or not at all. a) \(1 / 2 \int_{0+}(\log 1 / x)^{n} d x\) for \(-\infty

a) Define uniform convergence for improper integrals. b) Show that \({ }^{\infty} \int_{0} \mathrm{e}^{-\mathrm{xt}}\) dt converges uniformly to \(1 / \mathrm{x}\) in the interval \(1 \leq \mathrm{x} \leq 2\) c) Show that \(^{\infty} \int_{0} \mathrm{xe}^{-\mathrm{xt}}\) does not converge uniformly in the interval \(0 \leq \mathrm{x} \leq 1\)

Using the Weierstrass M-test, show that a) \(^{\infty} \int_{1}\left[(\cos x t) /\left(1+t^{2}\right)\right] d x\) converges uniformly in the interval \(\mathrm{A} \leq \mathrm{x} \leq \mathrm{B}\) b) \(1 \int_{0}^{+}[\\{\sin (t / x)\\} / \sqrt{t}]\) dt converges uniformly in any interval \(0<\mathrm{A} \leq \mathrm{x} \leq \mathrm{B}\)

Show that: a) \(\mathrm{F}(\mathrm{x})={ }^{\infty} \int_{0} \mathrm{e}^{-\mathrm{x}(\mathrm{t}) 2} \mathrm{dt}\) is continuous for \(\mathrm{x} \geq 1\) b) \((\partial / \partial \mathrm{x})^{\infty} \int_{0} \mathrm{e}^{-\mathrm{x}(\mathrm{t}) 2} \mathrm{dt}=-{ }^{\infty} \int_{0} \mathrm{t}^{2} \mathrm{e}^{-\mathrm{x}(\mathrm{t}) 2} \mathrm{dt}\) for \(\mathrm{x} \geq 1\). c) \(\lim _{\mathrm{x} \rightarrow 0}^{+\infty} \int_{0} \mathrm{xe}^{-\mathrm{xt}} \mathrm{dt} \neq{ }^{\infty} \int_{0}\left(\lim _{\mathrm{x} \rightarrow 0}^{+} \mathrm{xe}^{-\mathrm{xt}}\right) \mathrm{dt}\) and explain the result.

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 \(0<\mathrm{x}<1,0<\mathrm{y}<1\)

Show that a) if \(0<\mathrm{a}<\mathrm{b},{ }^{\infty} \int_{0}\left[\left(\mathrm{e}^{-\mathrm{at}}-\mathrm{e}^{-\mathrm{bt}}\right) / \mathrm{t}\right] \mathrm{dt}=\operatorname{In} \mathrm{b} / \mathrm{a}\). b) if \(\mathrm{a}>0,{ }^{\infty} \int_{0}\left[\left(\mathrm{e}^{-\mathrm{at}} \sin \mathrm{xt}\right) / \mathrm{t}\right] \mathrm{dt}=\tan ^{-1} \mathrm{x} / \mathrm{a}\).

Show that a) \(^{\infty} \int_{0} x^{-1} \sin x d x=(1 / 2) \pi\) (1) b) \(^{\infty} \int_{0}\left[\left(e^{-a x}-e^{-b x}\right) /(x \sec r x)\right] d x\) \(=(1 / 2) \operatorname{In}\left[\left(b^{2}+r^{2}\right) /\left(a^{2}+r^{2}\right)\right]\) where \(\mathrm{a}, \mathrm{b}>0\).

Show that \({ }^{\infty} \int_{0}[(\sin \mathrm{t}) \sqrt{t}] \mathrm{dt}=(\pi / 2)^{1 / 2}\) given that \({ }^{\infty} \int_{0} \mathrm{e}^{-(\mathrm{x}) 2} \mathrm{dx}(\sqrt{\pi} / 2)\), and that \({ }^{\infty} \int_{0}\left[\mathrm{ds} /\left(1+\mathrm{s}^{4}\right)\right]=[\pi /(2 \sqrt{2})]\).

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).

a) Prove that \(\Gamma(\mathrm{p}+1)=\mathrm{p} \Gamma(\mathrm{p}) \mathrm{p}>0\). b) show that \(\Gamma(p+k+1)=(p+k)(p+k-1) \ldots(p+2)(p+1) \Gamma(p+1)\)

By substituting \(\mathrm{y}=\mathrm{e}^{-\mathrm{x}}\) in the expression: $$ \Gamma(\mathrm{n})={ }^{\infty} \int_{0} \mathrm{x}^{\mathrm{n}-1} \mathrm{e}^{-\mathrm{x}} \mathrm{dx}(\mathrm{n}>0) $$ Obtain another form of \(\Gamma(\mathrm{n})\).

Verify the relation $$ \mathrm{B}(\mathrm{x}, \mathrm{y})=[\\{\Gamma(\mathrm{x}) \Gamma(\mathrm{y})\\} /\\{\Gamma(\mathrm{x}+\mathrm{y})\\}], \mathrm{x}, \mathrm{y}>0 $$ Where \(\mathrm{B}(\mathrm{x}, \mathrm{y})\) is the beta function and \(\Gamma(\mathrm{x})\) is the gamma function of \(\mathrm{x}\).

Prove \(\pi / 2 \int_{0} \sin ^{p} \theta \mathrm{d} \theta=\pi / 2 \int_{0} \cos ^{p} \theta \mathrm{d} \theta\) a) \(=[\\{1 \cdot 3 \cdot 5 \ldots(p-1)\\} /(2 \cdot 4 \cdot 6 \ldots p)]\) if \(p\) is an even Positive integer. b) \(=[\\{2 \cdot 4 \cdot 6 \ldots(p-1)\\} /(1 \cdot 3 \cdot 5 \ldots p)]\) if \(P\) is an odd positive integer. c) Evaluate \(\pi / 2 \int_{0} \cos ^{6} \theta \mathrm{d} \theta\).

Find the expression, in terms of \(\mathrm{n}\), for $$ 1 \int_{0}\left[\mathrm{dx} / \sqrt{\left.\left(1-\mathrm{x}^{\mathrm{n}}\right)\right]}\right. $$ Evaluate the result for \(\mathrm{n}=6\)