fresh_42 said:
7. a) Determine ##\int_1^\infty \frac{\log(x)}{x^3}\,dx\,.##
7. b) Determine for which ##\alpha## the integral ##\int_0^\infty x^2\exp(-\alpha x)\,dx## converges.
7. c) Find a sequence of functions ##f_n\, : \,\mathbb{R}\longrightarrow \mathbb{R}\, , \,n\in \mathbb{N}## such that $$\sum_\mathbb{N}\int_\mathbb{R}f_n(x)\,dx \neq \int_\mathbb{R}\left(\sum_\mathbb{N}f_n(x) \right) \,dx$$
7. d) Find a family of functions ##f_r\, : \,\mathbb{R}^+\longrightarrow \mathbb{R}\, , \,r\in \mathbb{R}## such that
$$
\lim_{r \to 0}\int_\mathbb{R}f_r(x)\,dx \neq \int_\mathbb{R} \lim_{r \to 0} f_r(x) \,dx
$$
7. e) Find an example for which
$$
\dfrac{d}{dx}\int_\mathbb{R}f(x,y)\,dy \neq \int_\mathbb{R}\dfrac{\partial}{\partial x}f(x,y)\,dy
$$
(by
@fresh_42 )
7. a) Let ##I_a:=\int_1^\infty \frac{\log(x)}{x^3}\,dx##. Let ##x=e^u\Rightarrow dx=e^u du## hence
$$I_a=\int_0^\infty ue^{-3u}e^u \, du=\int_0^\infty ue^{-2u}\, du=\int_0^\infty \tfrac{v}{2}e^{-v}\, \tfrac{dv}{2}=\tfrac{1}{4}\Gamma (2)=\boxed{\tfrac{1}{4}}$$
7. b) Determine for which ##\alpha## the integral ##\int_0^\infty x^2\exp(-\alpha x)\,dx## converges.
Let ##I_{\alpha}:=\int_0^\infty x^2e^{-\alpha x}\,dx##. For ##\alpha## positive, let ##u=\alpha x##, then
$$I_{\alpha}=\int_0^\infty \left( \tfrac{u}{\alpha}\right) ^2e^{-u}\,\tfrac{du}{\alpha}=\tfrac{1}{\alpha ^3}\int_0^\infty u^2e^{-u}\,du=\tfrac{1}{\alpha ^3}\Gamma (3)=\tfrac{2}{\alpha ^3}$$
Hence ##I_{\alpha}## converges for positive ##\alpha##. ##I_{\alpha}## obviously diverges for ##\alpha## non-positive. Let ##\alpha :=a+b i## then ##\left| I_{a+b i}\right| \leq\tfrac{2}{a^3}## whenever ##a>0##. Hence ##I_{\alpha}## converges for ##\Re \left[\alpha \right] >0##.
7. c) Find a sequence of functions ##f_n\, : \,\mathbb{R}\longrightarrow \mathbb{R}\, , \,n\in \mathbb{N}## such that $$\sum_\mathbb{N}\int_\mathbb{R}f_n(x)\,dx \neq \int_\mathbb{R}\left(\sum_\mathbb{N}f_n(x) \right) \,dx$$f
Define ##f_0(x):=\begin{cases}xe^{-\tfrac{1}{2}x^2}& \text{if } x\geq 0 \\0 & \text{ otherwise }\end{cases}##
and
##f_n (x) := \begin{cases} x e^{-\tfrac{1}{2} x^2} + \sum_{k=2}^{n} \left[ x k e^{-\tfrac{1}{2} k x^2}-x (k-1) e^{-\tfrac{1}{2} (k-1) x^2}\right] & \text{ if } x\geq 0 \\0 & \text{ otherwise} \end{cases}##
Aside: Let ##|a|<1##. Consider the sequence ##\left\{ na^n\right\}##: this sequence trivially converges to zero if ##a=0##. For ##0<|a|<1##, we may write, with ##\theta >0##
$$|a|=\tfrac{1}{1+\theta}\Rightarrow |a^n|=\tfrac{1}{1+\binom{n}{1}\theta +\cdots +\binom{n}{n}\theta ^n}$$
so we have for ##n=1, 2, 3,\ldots##,
$$|a^n|<\tfrac{1}{1+\binom{n}{2}\theta ^2}\Rightarrow |na^n|<\tfrac{1\cdot 2}{(n-1)\theta ^2}$$
Thus we have ##|na^n|<\epsilon ,## as soon as, ##\tfrac{1\cdot 2}{(n-1)\theta ^2}< \epsilon##
i.e. for every $$n>1+\tfrac{2}{\epsilon \cdot \theta ^2}$$
Hence ##na^n\rightarrow 0## for ##|a|<1##. End Aside.
Back to our sequence of functions ##\left\{ f_n\right\}##: Let ##s_n(x):=\sum_{k=1}^n f_k(x) = xne^{-\tfrac{1}{2}nx^2}##
$$\sum_\mathbb{N}\int_\mathbb{R}f_n(x)\,dx$$
$$=\sum_{n=1}^\infty \int_{0}^\infty\left\{ xe^{-\tfrac{1}{2}x^2} +x\sum_{k=2}^{n} \left[ ke^{-\tfrac{1}{2}kx^2}-(k-1)e^{-\tfrac{1}{2}(k-1)x^2}\right] \right\} \,dx$$
$$=\sum_{n=1}^\infty\left\{ \int_{0}^\infty xe^{-\tfrac{1}{2}x^2}dx +\sum_{k=2}^{n} \left[ \int_{0}^{\infty} xke^{-\tfrac{1}{2}kx^2}dx-\int_{0}^\infty x(k-1)e^{-\tfrac{1}{2}(k-1)x^2}dx\right] \right\}$$
$$=\sum_{n=1}^\infty\left\{ \tfrac{1}{2}\int_{0}^\infty e^{-u_1}du_1 +\sum_{k=2}^{n} \left[ \int_{0}^{\infty} e^{-u_2}du_2-\int_{0}^\infty e^{-u_3}du_3\right] \right\}$$
$$=\sum_{n=1}^\infty\left\{ \tfrac{1}{2}\Gamma (1) +\sum_{k=2}^{n} \left[ \Gamma(1)-\Gamma (1)\right] \right\} =\tfrac{1}{2}\sum_{n=1}^\infty \Gamma (1) \rightarrow +\infty$$
By the aside with ##a=e^{-\tfrac{1}{2}x^2}<1\forall x>0##, we have
##\int_\mathbb{R}\left(\sum_\mathbb{N}f_n(x) \right) \,dx=\int_\mathbb{R}\left(\lim_{n\to\infty}s_n(x) \right) \,dx=\int_\mathbb{R} 0 \, dx=0##
hence
$$\sum_\mathbb{N}\int_\mathbb{R}f_n(x)\,dx \neq \int_\mathbb{R}\left(\sum_\mathbb{N}f_n(x) \right) \,dx$$
7. d) Find a family of functions ##f_r\, : \,\mathbb{R}^+\longrightarrow \mathbb{R}\, , \,r\in \mathbb{R}## such that
$$\lim_{r \to 0}\int_\mathbb{R}f_r(x)\,dx\neq \int_\mathbb{R} \lim_{r \to 0} f_r(x) \,dx$$
Define ##f_r(x):=\begin{cases}\left\|\tfrac{1}{r}\right\| x(1-x^2)^{\left\|\tfrac{1}{r}\right\| }& \text{if } 0\leq x \leq 1 \\0 & \text{ otherwise } \end{cases}##
where ##\left\| y\right\|:=\text{nint}(y)##. Let n be the integer ##n:=\left\| \tfrac{1}{r}\right\|##,
$$\lim_{r \to 0}\int_\mathbb{R}f_r(x)\,dx=\lim_{r \to 0}\int_{0}^{1}\left\|\tfrac{1}{r}\right\| x(1-x^2)^{\left\|\tfrac{1}{r}\right\| }\, dx$$
$$ =\lim_{n \to \infty}\tfrac{n}{2}\int_{0}^{1}u^{n}\, du =\lim_{n \to \infty}\tfrac{n}{2(n+1)}=\tfrac{1}{2} $$
but
$$ \int_{\mathbb{R}} \lim_{r \to 0} f_r(x) \, dx = \int_{0}^{1} \lim_{r \to 0} \left\| \tfrac{1}{r} \right\| x(1-x^2)^{ \left\| \tfrac{1}{r} \right\|} \, dx $$
$$ = \int_{0}^{1} \lim_{n \to\infty} n x(1-x^2)^{n}\,dx =0 $$
where the limit was evaluated by the aside to problem 7 c). Hence
$$\lim_{r \to 0}\int_\mathbb{R}f_r(x)\,dx\neq \int_\mathbb{R} \lim_{r \to 0} f_r(x) \,dx$$
7. e) Find an example for which
$$\dfrac{d}{dx}\int_\mathbb{R}f(x,y)\,dy \neq \int_\mathbb{R}\dfrac{\partial}{\partial x}f(x,y)\,dy$$
This is baby Rudin pg. 242 ch 9 exercise #28:
Let ##f(x,y)=\begin{cases} y-2\sqrt{-x} & \text{if } \sqrt{|x|}\leq y \leq 2\sqrt{|x|} \wedge x<0 \\ -y & \text{if } 0 \leq y \leq \sqrt{|x|} \wedge x< 0 \\ y & \text{if } 0 \leq y \leq \sqrt{x} \wedge x\geq 0 \\ -y+2\sqrt{x} & \text{if } \sqrt{x}\leq y \leq 2\sqrt{x} \wedge x\geq 0 \\ 0 & \text{ otherwise} \end{cases}##
Let ##g(x):=\int_{\mathbb{R}}f(x,y)\, dy=\begin{cases} \int_{\sqrt{-x}}^{2\sqrt{-x}}( y-2\sqrt{-x})\, dy-\int_{0}^{\sqrt{-x}}y\, dy & \text{if } x<0 \\ \int_{0}^{\sqrt{x}}y\, dy+\int_{\sqrt{x}}^{2\sqrt{x}}(-y+2\sqrt{x})\, dy & \text{if } x\geq 0 \end{cases}##
Recall the formula for differentiating under the integral sign:
$$\boxed{\frac{d}{dx}\int_{a(x)}^{b(x)} h(x,y)\, dy =\int_{a(x)}^{b(x)} \tfrac{\partial h}{\partial x}\, dy + h(x,b(x)\tfrac{db}{dx}-h(x,a(x)\tfrac{da}{dx}}$$
Then, differentiating under the integral sign, we have
$$g^{\prime}(x)=
\begin{cases} \int_{\sqrt{-x}}^{2\sqrt{-x}}\left( y+\tfrac{1}{\sqrt{-x}}\right)\, dy+0-(-\sqrt{-x})\left( -\tfrac{1}{\sqrt{-x}}\right) -0-(\sqrt{-x})\left( -\tfrac{1}{2\sqrt{-x}}\right)+0 & \text{if } x<0 \\ 0+(\sqrt{x})\left( \tfrac{1}{2\sqrt{x}}\right) -0+\int_{\sqrt{x}}^{2\sqrt{x}}\left( -y+\tfrac{1}{\sqrt{x}}\right) \, dy +0-(\sqrt{x})\left( \tfrac{1}{2\sqrt{x}}\right) & \text{if } x\geq 0 \end{cases}$$
cc
$$\Rightarrow g^{\prime}(x)=\begin{cases} \left[ \tfrac{1}{2}y^2+\tfrac{y}{\sqrt{-x}}\right|_{y=\sqrt{-x}}^{2\sqrt{-x}} & \text{if } x<0 \\ \left[ -\tfrac{1}{2}y^2+\tfrac{y}{\sqrt{x}}\right|_{y=\sqrt{x}}^{2\sqrt{x}}& \text{if } x\geq 0 \end{cases}=\begin{cases} \tfrac{3}{2}|x|+1 & \text{if } x<0 \\ -\tfrac{3}{2}x+1& \text{if } x\geq 0 \end{cases}=1-\tfrac{3}{2}x$$
$$\Rightarrow \boxed{g^{\prime}(x)=1-\tfrac{3}{2}x, \, \, \forall x\in\mathbb{R}}$$
Now ##f_{x}(x,y)=\begin{cases} \tfrac{1}{\sqrt{-x}} & \text{if } \sqrt{|x|}\leq y \leq 2\sqrt{|x|} \wedge x<0 \\ 0 & \text{if } 0 \leq y \leq \sqrt{|x|} \wedge x< 0 \\ 0 & \text{if } 0 \leq y \leq \sqrt{x} \wedge x\geq 0 \\ \tfrac{1}{\sqrt{x}} & \text{if } \sqrt{x}\leq y \leq 2\sqrt{x} \wedge x\geq 0 \\ 0 & \text{ otherwise} \end{cases}##
Hence
$$F(x):=\int_{\mathbb{R}}f_{x}(x,y)\, dy =\begin{cases} \int_{\sqrt{-x}}^{\sqrt{-x}}\tfrac{dy}{\sqrt{-x}}+0 & \text{if } x<0 \\ 0+ \int_{\sqrt{x}}^{\sqrt{x}}\tfrac{dy}{\sqrt{x}} & \text{if } x<0 \\ 0 & \text{if } x=0 \end{cases}$$
$$\boxed{F(x)=\begin{cases} 1 & \text{if } |x|>0 \\ 0 & \text{if } x=0 \end{cases}}$$
thus indeed
$$F(x):=\int_{\mathbb{R}}f_{x}(x,y)\, dy \neq \tfrac{d}{dx}\int_{\mathbb{R}}f(x,y)\, dy =: g^{\prime}(x)$$