Chapter V

The Integral

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Section V.1Improper Integrals of Functions of One Variable[47/100%]
Problem V.1.1

Investigate the convergence of the integral ∫1∞∣sin⁡x∣dxexp⁡(x2sin⁡2x)\int_{1}^{\infty } \frac{|\sin x| d x}{\exp \left(x^{2} \sin^{2} x\right)}.

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Problem V.1.2

Investigate the convergence of the integral ∫1∞dx1+xpsin⁡2x\int_{1}^{\infty } \frac{d x}{1+x^{p} \sin^{2} x}.

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Problem V.1.3

Investigate the convergence of the integral ∫1∞xqdx1+xp∣sin⁡x∣r\int_{1}^{\infty } \frac{x^{q} d x}{1+x^{p}|\sin x|^{r}}.

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Problem V.1.4

Investigate the convergence of the integral ∫1∞xqdxexp⁡(xp∣sin⁡x∣r)\int_{1}^{\infty } \frac{x^{q} d x}{\exp \left(x^{p}|\sin x|^{r}\right)}.

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Problem V.1.5

Investigate the convergence of the integral ∫1∞∣sin⁡xp∣xqdx\int_{1}^{\infty } \frac{\left|\sin x^{p}\right|}{x^{q}} d x.

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Problem V.1.6

Investigate the convergence of the integral ∫1∞∣sin⁡(xp+ln⁡qx)∣dxx\int_{1}^{\infty }\left|\sin \left(x^{p}+\ln^{q} x\right)\right| \frac{d x}{x}.

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Problem V.1.7

Investigate the convergence of the integral ∫1∞xp∣sin⁡x∣xqdx\int_{1}^{\infty } x^{p}|\sin x|^{x^{q}} d x.

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Problem V.1.8

Investigate the convergence of the integral ∫1∞cos⁡(x3−x)dx\int_{1}^{\infty } \cos \left(x^{3}-x\right) d x.

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Problem V.1.9

Investigate the convergence of the integral ∫1∞sin⁡(xln⁡x)dx\int_{1}^{\infty } \sin (x \ln x) d x.

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Problem V.1.10

Investigate the convergence of the integral ∫1∞sin⁡(xp+ax+b)dx,p>1\int_{1}^{\infty } \sin \left(x^{p}+a x+b\right) d x, p>1.

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Problem V.1.11

Investigate the convergence of the integral ∫π∞sin⁡xdxln⁡x+cos⁡x\int_{\pi }^{\infty } \frac{\sin x d x}{\ln x+\cos \sqrt{x}}.

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Problem V.1.12

Investigate the convergence of the integral ∫π∞sin⁡xdxln⁡x+cos⁡x\int_{\pi }^{\infty } \frac{\sin x d x}{\ln x+\cos x}.

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Problem V.1.13

Investigate the convergence of the integral ∫2π∞sin⁡xdxln⁡x+cos⁡x+sin⁡x\int_{2 \pi }^{\infty } \frac{\sin x d x}{\ln x+\cos x+\sin x}.

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Problem V.1.14

Investigate the convergence of the integral ∫π∞sin⁡xdxln⁡x+cos⁡x2\int_{\pi }^{\infty } \frac{\sin x d x}{\ln x+\cos x^{2}}.

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Problem V.1.15

Does the convergence of the integral ∫1∞f(x)dx\int_{1}^{\infty } f(x) d x imply the convergence of the integrals

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(a)

∫1∞f3(x)dx\int_{1}^{\infty } f^{3}(x) d x;

(b)

∫1∞∣f(x)∣dxx3\int_{1}^{\infty }\left|f(x)\right| \frac{d x}{x^{3}} ?

Problem V.1.16
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(a)

Prove that for a bounded function ff decreasing to zero as x→+∞x \rightarrow +\infty, the integrals

∫0∞∣sin⁡x∣f(x)dx and ∫0∞f(x)dx \int _{0}^{\infty }\left|\sin x\right| f(x) d x \quad \text{ and } \quad \int _{0}^{\infty } f(x) d x

converge or diverge simultaneously.

(b)

Verify that in the two-dimensional case the analogue of this statement is false:

∫0∞∫0∞∣sin⁡x∣∣sin⁡y∣exyln⁡(x+y+2)dxdy<+∞, but ∫0∞∫0∞dxdyexyln⁡(x+y+2)=+∞ \int _{0}^{\infty } \int _{0}^{\infty } \frac{\left|\sin x\right||\sin y|}{e^{x y} \ln (x+y+2)} d x d y<+\infty , \quad \text{ but } \quad \int _{0}^{\infty } \int _{0}^{\infty } \frac{d x d y}{e^{x y} \ln (x+y+2)}=+\infty
Problem V.1.17

Let

f(x)=1xε∫0xln⁡∣1−t2∣dtt2(x>0) f(x)=\frac{1}{x^{\varepsilon }} \int _{0}^{x} \ln \left|1-t^{2}\right| \frac{d t}{t^{2}} \quad (x>0)

For which ε>0\varepsilon >0 does the integral ∫0∞f(x)dx\int_{0}^{\infty } f(x) d x converge?

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Problem V.1.18

Compute the Frullani integral

∫0∞f(ax)−f(bx)xdx \int _{0}^{\infty } \frac{f(a x)-f(b x)}{x} d x

where a,b>0a, b>0, the function ff is continuous on [0,∞)[0, \infty ) and satisfies one of the conditions:

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(a)

the integral ∫1∞f(x)xdx\int_{1}^{\infty } \frac{f(x)}{x} d x converges;

(b)

for some T>0T>0, for all x⩾0x \geqslant 0 the equality f(x+T)=f(x)f(x+T)=f(x) holds;

(c)

there exists a finite limit l=lim⁡x→∞f(x)l=\lim_{x \rightarrow \infty } f(x).

Problem V.1.19

Suppose the integral ∫0∞f(x)dx\int_{0}^{\infty } f(x) d x converges. Prove that

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(a)

for any ε>0\varepsilon >0 the integral ∫0∞e−εxf(x)dx\int_{0}^{\infty } e^{-\varepsilon x} f(x) d x converges;

(b)

∫0∞e−εxf(x)dx→∫0∞f(x)dx\int_{0}^{\infty } e^{-\varepsilon x} f(x) d x \rightarrow \int_{0}^{\infty } f(x) d x as ε→+0\varepsilon \rightarrow +0 (compare with Abel's theorem — Problem IV.5.8).

Problem V.1.20

Let φ\varphi be a continuous 2π2 \pi-periodic function. Prove that the integral

∫π∞φ(x)dxln⁡x+cos⁡x \int _{\pi }^{\infty } \frac{\varphi (x) d x}{\ln x+\cos x}

converges only if φ\varphi is an odd function.

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Problem V.1.21

Compute the integral ∫0∞ln⁡∣1−x2∣dxx2\int_{0}^{\infty } \ln \left|1-x^{2}\right| \frac{d x}{x^{2}}.

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Problem V.1.22

Compute the integral ∫0π/2ln⁡(sin⁡x)dx\int_{0}^{\pi / 2} \ln (\sin x) d x.

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Problem V.1.23

Compute the integral ∫01ln⁡xdx1−x\int_{0}^{1} \frac{\ln x d x}{1-x}.

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Problem V.1.24

Compute the integral ∫01ln⁡xdx1+x\int_{0}^{1} \frac{\ln x d x}{1+x}.

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Problem V.1.25

Compute the integral ∫0∞x2ch⁡2xdx\int_{0}^{\infty } \frac{x^{2}}{\operatorname {ch}^{2} x} d x.

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Problem V.1.26

Compute the integral ∫0∞sin⁡xxe−axdx(a>0)\int_{0}^{\infty } \frac{\sin x}{x} e^{-a x} d x \quad (a>0).

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Problem V.1.27

Compute the integral ∫0∞sin⁡xxdx\int_{0}^{\infty } \frac{\sin x}{x} d x.

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Problem V.1.28

Compute the integral ∫0∞x−sin⁡xx3dx\int_{0}^{\infty } \frac{x-\sin x}{x^{3}} d x.

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Problem V.1.29

Compute the integral ∫0∞e−x−cos⁡xxdx\int_{0}^{\infty } \frac{e^{-x}-\cos x}{x} d x.

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Problem V.1.30

Compute the integral ∫−∞∞(eiax−1)(eibx−1)x2dx(a,b∈R)\int_{-\infty }^{\infty } \frac{\left(e^{i a x}-1\right)\left(e^{i b x}-1\right)}{x^{2}} d x \quad (a, b \in \mathbb {R}).

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Problem V.1.31

Compute the integral ∫0∞ln⁡xexdx\int_{0}^{\infty } \frac{\ln x}{e^{x}} d x.

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Remark.
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The answer is expressed in terms of Euler's constant γ\gamma (see Problem II.2.8).

Problem V.1.32

Compute the integral ∫0∞ln⁡2xexdx\int_{0}^{\infty } \frac{\ln^{2} x}{e^{x}} d x.

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Remark.
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The answer is expressed in terms of Euler's constant γ\gamma (see Problem II.2.8).

Problem V.1.33

Compute the integral ∫01x2dx(1−x)ln⁡2(1−x)\int_{0}^{1} \frac{x^{2} d x}{(1-x) \ln^{2}(1-x)}.

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Problem V.1.34

Compute the integral ∫0∞ln⁡x1+exdx\int_{0}^{\infty } \frac{\ln x}{1+e^{x}} d x.

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Problem V.1.35

Compute the integrals:

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(a)

∫01e−x−1xdx+∫1∞e−xxdx\int_{0}^{1} \frac{e^{-x}-1}{x} d x+\int_{1}^{\infty } \frac{e^{-x}}{x} d x;

(b)

∫01cos⁡x−1xdx+∫1∞cos⁡xxdx\int_{0}^{1} \frac{\cos x-1}{x} d x+\int_{1}^{\infty } \frac{\cos x}{x} d x.

Remark.
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The answer is expressed in terms of Euler's constant γ\gamma (see Problem II.2.8).

Problem V.1.36

Compute the integrals:

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(a)

∫0∞cos⁡(xp)−cos⁡(xq)xdx(p,q>0)\int_{0}^{\infty } \frac{\cos \left(x^{p}\right)-\cos \left(x^{q}\right)}{x} d x \quad (p, q>0);

(b)

∫0∞exp⁡(−xp)−exp⁡(−xq)xdx(p,q>0)\int_{0}^{\infty } \frac{\exp \left(-x^{p}\right)-\exp \left(-x^{q}\right)}{x} d x \quad (p, q>0).

Remark.
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The answer is expressed in terms of Euler's constant γ\gamma (see Problem II.2.8).

Problem V.1.37

Compute the integral ∫1∞{x}(1−{x})x2dx({x}\int_{1}^{\infty } \frac{\{ x\} (1-\{ x\} )}{x^{2}} d x \quad (\{ x\} — the fractional part of x)x).

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Problem V.1.38

Using the result of problem I.2.23 a), prove that

∫0ne−x2dx=n(2n)!!(2n+1)!!+o(1) \int _{0}^{\sqrt{n}} e^{-x^{2}} d x=\sqrt{n} \frac{(2 n)!!}{(2 n+1)!!}+o(1)

Using Stirling's formula, find the value of the Euler-Poisson integral ∫−∞∞e−x2dx\int_{-\infty }^{\infty } e^{-x^{2}} d x.

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Problem V.1.39

Compute the integral ∫0∞e−x−e−2xxpdx\int_{0}^{\infty } \frac{e^{-x}-e^{-2 x}}{x^{p}} d x, where p=k2−1,k=0,1,2,3,4,5p=\frac{k}{2}-1, k=0,1,2,3,4,5.

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Problem V.1.40

Compute the integral ∫0∞e−x2(x2+12)2dx\int_{0}^{\infty } \frac{e^{-x^{2}}}{\left(x^{2}+\frac{1}{2}\right)^{2}} d x.

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Problem V.1.41

Compute the integral ∫0∞e−(x2+a2x−2)dx\int_{0}^{\infty } e^{-\left(x^{2}+a^{2} x^{-2}\right)} d x.

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Problem V.1.42

Compute the integrals:

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(a)

∫0∞sin⁡x2dx\int_{0}^{\infty } \sin x^{2} d x;

(b)

∫0∞cos⁡x2dx\int_{0}^{\infty } \cos x^{2} d x (Fresnel integrals).

Problem V.1.43

Compute the integral ∫0∞1−∣cos⁡x∣px2dx(p>−1)\int_{0}^{\infty } \frac{1-|\cos x|^{p}}{x^{2}} d x \quad (p>-1).

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Problem V.1.44

Compute the integral ∫2∞π(x)x3−xdx\int_{2}^{\infty } \frac{\pi (x)}{x^{3}-x} d x, where π(x)\pi (x) is the number of primes not exceeding xx.

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Problem V.1.45

Let φ(x)=e2πix−1x,x∈R\varphi (x)=\frac{e^{2 \pi i x}-1}{x}, x \in \mathbb {R}. Prove that for any n∈Zn \in \mathbb {Z}, n≠0n \neq 0

∫−∞∞φ(x)φˉ(x−n)dx=0 \int _{-\infty }^{\infty } \varphi (x) \bar{\varphi }(x-n) d x=0

What is this integral equal to when n=0n=0?

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Problem V.1.46

Prove that ∫0∞xnsin⁡(2πln⁡x)dxxln⁡x=0\int_{0}^{\infty } x^{n} \sin (2 \pi \ln x) \frac{d x}{x^{\ln x}}=0 for n=0,1,2,…n=0,1,2, \ldots

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Problem V.1.47

Prove that ∫0π/2cos⁡(p+1)xcos⁡p−1xdx=0\int_{0}^{\pi / 2} \cos (p+1) x \cos^{p-1} x d x=0 for all p>0p>0.

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Section V.2Computing Multiple Integrals[19/100%]
Problem V.2.1

Evaluate the integral ∫0+∞∫0+∞∣ln⁡x−ln⁡y∣e−(x+y)dxdy\int_{0}^{+\infty } \int_{0}^{+\infty }|\ln x-\ln y| e^{-(x+y)} d x d y.

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Problem V.2.2

Evaluate the integrals

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(a)

∫(0,1)nmax⁡(x1,…,xn)dx1…dxn\int_{(0,1)^{n}} \max \left(x_{1}, \ldots , x_{n}\right) d x_{1} \ldots d x_{n};

(b)

∫(0,1)nmin⁡(x1,…,xn)dx1…dxn\int_{(0,1)^{n}} \min \left(x_{1}, \ldots , x_{n}\right) d x_{1} \ldots d x_{n};

(c)

∫(0,1)n(min⁡(x1,…,xn)max⁡(x1,…,xn))pdx1…dxn(p>−1)\int_{(0,1)^{n}}\left(\frac{\min \left(x_{1}, \ldots , x_{n}\right)}{\max \left(x_{1}, \ldots , x_{n}\right)}\right)^{p} d x_{1} \ldots d x_{n} \quad (p>-1);

(d)

∫[1,+∞)ndx1…dxnx1…xn(max⁡(x1,…,xn))ε(ε>0)\int_{[1,+\infty )^{n}} \frac{d x_{1} \ldots d x_{n}}{x_{1} \ldots x_{n}\left(\max \left(x_{1}, \ldots , x_{n}\right)\right)^{\varepsilon }} \quad (\varepsilon >0).

Problem V.2.3

Evaluate the integral ∫Rne−∥x∥2/2dx\int_{\mathbb {R}^{n}} e^{-\| x\|^{2} / 2} d x.

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Problem V.2.4

Evaluate the integrals

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(a)

∬R2∣ax+by∣dxdye(x2+y2)/2\iint_{\mathbb {R}^{2}}\left|a x+b y\right| \frac{d x d y}{e^{\left(x^{2}+y^{2}\right) / 2}};

(b)

∫Rn∣(a,x)∣pdxe∥x∥2/2(a∈Rn,p>−1)\int_{\mathbb {R}^{n}}\left|(a, x)\right|^{p} \frac{d x}{e^{\left\| x\right\|^{2} / 2}} \quad \left(a \in \mathbb {R}^{n}, p>-1\right).

Problem V.2.5

Evaluate the integral ∫∥y∥⩽1dy∥x−y∥(x,y∈R3)\int_{\| y\| \leqslant 1} \frac{d y}{\| x-y\| } \quad \left(x, y \in \mathbb {R}^{3}\right).

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Problem V.2.6

Evaluate the integral ∭∫x2+y2+u2+v2⩽1ex2+y2−u2−v2dxdydudv\iiint \int_{x^{2}+y^{2}+u^{2}+v^{2} \leqslant 1} e^{x^{2}+y^{2}-u^{2}-v^{2}} d x d y d u d v.

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Problem V.2.7

Compute the integral ∫(Ax,x)⩽1e(Ax,x)dx\int_{(A x, x) \leqslant 1} e^{(A x, x)} d x (x∈R4,A\left(x \in \mathbb {R}^{4}, A\right. is a positive definite matrix).

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Problem V.2.8

Let A={(x1,x2,x3,x4)∈R4∣x22+x32+x42<x1}A=\left\{ \left(x_{1}, x_{2}, x_{3}, x_{4}\right) \in \mathbb {R}^{4} \mid \sqrt{x_{2}^{2}+x_{3}^{2}+x_{4}^{2}}<x_{1}\right\}. For which t∈R4t \in \mathbb {R}^{4} is the integral

K(t)=∫Ae−(x,t)dx K(t)=\int _{A} e^{-(x, t)} d x

finite, and what is it equal to?

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Problem V.2.9

On the segment [a,b]\left[a, b\right] two points are chosen at random. Find the mean value MM of the distance between them. What is the probability PP that the distance is greater than MM? For which mm are the events "the distance is greater than mm" and "the distance is less than mm" equally probable?

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Remark.
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When speaking of a random choice of points, we mean that the probability of choosing a point belonging to some set is proportional to its measure (length, area, volume).

Problem V.2.10

On the segment [0,a]\left[0, a\right] three numbers are chosen at random. What is the probability that they are the lengths of the sides of some triangle?

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Remark.
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When speaking of a random choice of points, we mean that the probability of choosing a point belonging to some set is proportional to its measure (length, area, volume).

Problem V.2.11

On the segment [−a,a]\left[-a, a\right] two points u,vu, v are chosen at random.

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(a)

Which is more probable: that the roots of the equation z2+uz+v=0z^{2}+u z+v=0 lie on the real axis (probability P1(a)P_{1}(a)) or that the roots of this equation do not lie on the real axis (probability P2(a)P_{2}(a))? What do the probabilities P1(a),P2(a)P_{1}(a), P_{2}(a) tend to as a→+∞a \rightarrow +\infty?

(b)

What is the probability that the biquadratic equation z4+uz2+v=0z^{4}+u z^{2}+v=0 has both real and complex roots?

Remark.
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When speaking of a random choice of points, we mean that the probability of choosing a point belonging to some set is proportional to its measure (length, area, volume).

Problem V.2.12

In the unit disk two points are chosen at random. What is the mean value SS of the area of the circle constructed on the segment joining them as diameter?

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Remark.
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When speaking of a random choice of points, we mean that the probability of choosing a point belonging to some set is proportional to its measure (length, area, volume).

Problem V.2.13

On a circle of radius RR two points are chosen at random. Find the mean value LL of the length of the chord joining them.

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Remark.
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When speaking of a random choice of points, we mean that the probability of choosing a point belonging to some set is proportional to its measure (length, area, volume).

Problem V.2.14

Let

A={(x1,…,xn)∈Rn∣ ∑1⩽k⩽nxkk⩽1,x1,…,xn⩾0}. A=\left\{ \left(x_{1}, \ldots , x_{n}\right) \in \mathbb {R}^{n} \left\lvert \, \sum _{1 \leqslant k \leqslant n} \frac{x_{k}}{k} \leqslant 1\right., x_{1}, \ldots , x_{n} \geqslant 0\right\} .

Prove that for any t∈Rt \in \mathbb {R} the equality

∫…∫et(x1+…+xn)dx1…dxn=(et−1t)n \int \ldots \int e^{t\left(x_{1}+\ldots +x_{n}\right)} d x_{1} \ldots d x_{n}=\left(\frac{e^{t}-1}{t}\right)^{n}

holds.

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Problem V.2.15

Prove that for m,n∈N,m⩽n,a>0m, n \in \mathbb {N}, m \leqslant n, a>0, the equality

∫[0,1]ndx(a+x1+…+xn)m=(−1)m−1Δn(an−mln⁡a)(m−1)!(n−m)! \int _{\left[0,1\right]^{n}} \frac{d x}{\left(a+x_{1}+\ldots +x_{n}\right)^{m}}=(-1)^{m-1} \frac{\Delta _{n}\left(a^{n-m} \ln a\right)}{(m-1)!(n-m)!}

holds, where Δ1f(a)=Δf(a)=f(a+1)−f(a),Δnf(a)=Δ(Δn−1f(a))\Delta_{1} f(a)=\Delta f(a)=f(a+1)-f(a), \quad \Delta_{n} f(a)=\Delta \left(\Delta_{n-1} f(a)\right).

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Problem V.2.16

Let a0,…,ana_{0}, \ldots , a_{n} be positive numbers, b0=a1b_{0}=a_{1}, bk=ak+1−akb_{k}=a_{k+1}-a_{k} for k=1,…,n(an+1=a0)k=1, \ldots , n\left(a_{n+1}=a_{0}\right),

Wn={(x1,…,xn)∈Rn∣∑1⩽k⩽nxk<1,x1,…,xn>0}. W_{n}=\left\{ \left(x_{1}, \ldots , x_{n}\right) \in \mathbb {R}^{n} \mid \sum _{1 \leqslant k \leqslant n} x_{k}<1, x_{1}, \ldots , x_{n}>0\right\} .
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(a)

Prove that for any nonnegative and continuous function ff on (0,+∞)(0,+\infty ) the equality

∫…∫f(b0+∑k=1nbkx1…xk)x1n−1x2n−2…xn−1dx1…dxn==∫…∫f(a0+∑k=1n(ak−a0)xk)dx1…dxn \begin{aligned} \int \ldots \int f\left(b_{0}+\sum _{k=1}^{n} b_{k} x_{1} \ldots x_{k}\right) x_{1}^{n-1} x_{2}^{n-2} \ldots x_{n-1} d x_{1} \ldots d x_{n} & = \\ & =\int \ldots \int f\left(a_{0}+\sum _{k=1}^{n}\left(a_{k}-a_{0}\right) x_{k}\right) d x_{1} \ldots d x_{n} \end{aligned}

holds.

(b)

Compute the integrals

∫…∫(0,1)nx1n−1x2n−2…xn−1dx1…dxn(b0+b1x1+b2x1x2+…+bnx1…xn)n+1∫…∫dx1…dxn(a0+(a1−a0)x1+(a2−a0)x2+…+(an−a0)xn)n+1. \begin{gathered} \int \ldots \int _{(0,1)^{n}} \frac{x_{1}^{n-1} x_{2}^{n-2} \ldots x_{n-1} d x_{1} \ldots d x_{n}}{\left(b_{0}+b_{1} x_{1}+b_{2} x_{1} x_{2}+\ldots +b_{n} x_{1} \ldots x_{n}\right)^{n+1}} \\ \int \ldots \int \frac{d x_{1} \ldots d x_{n}}{\left(a_{0}+\left(a_{1}-a_{0}\right) x_{1}+\left(a_{2}-a_{0}\right) x_{2}+\ldots +\left(a_{n}-a_{0}\right) x_{n}\right)^{n+1}} . \end{gathered}
Problem V.2.17

Prove that

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(a)

if the function ff is continuous and nonnegative on R\mathbb {R}, then for any real numbers aa and bb the equality

1π∬R2f(ax+by)dxdy(1+x2)(1+y2)=∫−∞+∞f((∣a∣+∣b∣)u)du1+u2 \frac{1}{\pi } \iint _{\mathbb {R}^{2}} \frac{f(a x+b y) d x d y}{\left(1+x^{2}\right)\left(1+y^{2}\right)}=\int _{-\infty }^{+\infty } f((\left|a\right|+\left|b\right|) u) \frac{d u}{1+u^{2}}

holds

(b)

for 0<p<10<p<1 and a=(a1,…,an)∈Rna=\left(a_{1}, \ldots , a_{n}\right) \in \mathbb {R}^{n} the equality (x=(x1,…,xn)∈Rn)\left(x=\left(x_{1}, \ldots , x_{n}\right) \in \mathbb {R}^{n}\right)

∫Rn∣(a,x)∣pdx1…dxn(1+x12)(1+x22)…(1+xn2)=πncos⁡π2p(∑1⩽k⩽n∣ak∣)p. \int _{\mathbb {R}^{n}} \frac{\left|(a, x)\right|^{p} d x_{1} \ldots d x_{n}}{\left(1+x_{1}^{2}\right)\left(1+x_{2}^{2}\right) \ldots \left(1+x_{n}^{2}\right)}=\frac{\pi ^{n}}{\cos \frac{\pi }{2} p}\left(\sum _{1 \leqslant k \leqslant n}\left|a_{k}\right|\right)^{p} .

holds.

Problem V.2.18

Let f∈C([0,1]2)f \in C\left(\left[0,1\right]^{2}\right). Find the limit

lim⁡n→∞n2(∬[0,1]2f(x,y)dxdy−1n2∑1⩽k⩽n1⩽j⩽nf(k−12n,j−12n)) \lim _{n \rightarrow \infty } n^{2}\left(\iint _{\left[0,1\right]^{2}} f(x, y) d x d y-\frac{1}{n^{2}} \sum _{\substack {1 \leqslant k \leqslant n \\ 1 \leqslant j \leqslant n}} f\left(\frac{k-\frac{1}{2}}{n}, \frac{j-\frac{1}{2}}{n}\right)\right)
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Problem V.2.19

Let p<1p<1 and

Φ(x,y)=∬u2+v2⩽1((x−u)2+(y−v)2)−p/2dudvΨ(x,y)=∬u2+v2⩽1ln⁡((x−u)2+(y−v)2)dudv \begin{aligned} & \Phi (x, y)=\iint _{u^{2}+v^{2} \leqslant 1}\left((x-u)^{2}+(y-v)^{2}\right)^{-p / 2} d u d v \\ & \Psi (x, y)=\iint _{u^{2}+v^{2} \leqslant 1} \ln \left((x-u)^{2}+(y-v)^{2}\right) d u d v \end{aligned}

Prove that Φ,Ψ∈C1(R2)\Phi , \Psi \in C^{1}\left(\mathbb {R}^{2}\right), and find Ψ\Psi.

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