1
vote
1answer
12 views

Let $\int_a^bf(x)sgn(f(x)) + 2f(x) \ dx = 0$. Show that $f$ has at least one root.

The Assignment: Let $a,b \in\mathbb{R}$ and $a < b$. Furthermore let $f: [a,b] \rightarrow \mathbb{R}$ be differentiable and $|f(x)| + |f'(x)| \neq 0$ for $\forall x \in [a,b]$. Now, let ...
0
votes
1answer
25 views

Limits of integration

Is there any difference between the $$\int_a^b f(x) dx $$ and $$\lim_{x\to b^-} \int_a^x f(x) dx \qquad \text{OR}\qquad \lim_{x\to a^+} \int_x^b f(x) dx$$ When would one need the second versions of ...
0
votes
1answer
13 views

Showing that this function is not riemann integrable.

Consider the function h defined by h(x) := x+1 for x an element of [0,1] rational, and h(x) := 0 for x an element of [0,1] irrational. Show that h is not Riemann integrable. The hint in the back of ...
2
votes
2answers
18 views

Finding an integral $\int g(x)^j dx $ from $\int g(x)^2 dx $

let $I = \int_0^1 g(x)^2 dx $, where $g$ is a real valued function. With this information is it possible to give an upper bound for $\int_0^1 g(x)^j dx $? Here $j$ is a natural number. When $j=1$ I ...
3
votes
2answers
43 views

How to show that $f$ is Riemann integrable

Let $u,v:[a,b]\rightarrow\mathbb{R}$ be contunious. Define $f:[a,b]\rightarrow\mathbb{R}$ by $$f(x) = \begin{cases}u(x) & x \in \mathbb{Q} \\ v(x) & x \in \mathbb{R}-\mathbb{Q}\end{cases}$$ ...
2
votes
0answers
52 views

How to integrate $\displaystyle\int_o^\pi\frac{dx}{\sqrt{3-\cos(x)}}$?

How to integrate $\displaystyle\int_o^\pi\frac{dx}{\sqrt{3-\cos(x)}}$ ? If I take $y=\sin\left(\frac{x}{2}\right)$ then, $\displaystyle ...
0
votes
1answer
33 views

Support of $L^p$ functions?

I noticed something strange. If we look at a function $f \in L^p$, then this is an equivalence class. Hypothetically: $\operatorname{supp}(f) = \overline{\{f\neq 0\}}$. But this is strange, as $f$ is ...
0
votes
0answers
27 views

Generate Borel Sigma Algebra

I want to show that the Borel Sigma-Algebra on $\mathbb{R}^n$ is generated by $ A:= \{(a_1,b_1] \times \cdots\times (a_n,b_n]; a_i,b_i \in \mathbb{R} \}$ as well as $ B:= \{(-\infty,c_1] ...
-2
votes
0answers
24 views

change of variable in the integral with measure $\mu$ [on hold]

Let $f:(X,\mathbb{X}, \mu) \to Y$ a measurable function and $\nu(A)=\mu(f^{-1}(A))$. show that $$\int_Yg \,d\nu = \int_X g(f(x)) \,d\mu(x)$$
2
votes
4answers
70 views

Evaluating the improper integral $\int_0^{\infty} \frac{\sin x}{x+x^2} \ dx$

Evaluating the improper integral $$\int_0^{\infty} \frac{\sin x}{x+x^2} \ dx$$ I'm trying to determine if the integral exists. I can't seem to deal with $$\lim_{a\to 0^+} \int_a^\infty \frac{\sin ...
0
votes
0answers
13 views

Mehler-Sonine integrals for $Y_{\nu}(x)$ Bessel function

My question is regarding the following integral representation of the $Y_{\nu}(x)$ Bessel function (Watson page 170): $$ Y_{\nu}(x) = \frac{2 (\frac{1}{2} x)^{-\nu}}{{\pi}^{1/2} \Gamma(\frac{1}{2} - ...
0
votes
1answer
37 views

L1 convergence and Lp bounded implies Lq convergence

I have tried to solve this problem for almost a week and did not manage to, so I figured to ask it here: Let $(u_n)\to u$ in $L^1(0,1)$ strongly and let $\{u_n\}_{n\in\mathbb{N}}$ be bounded in ...
2
votes
1answer
31 views

a geometric interpretation of a line integral

Is there a geometric interpretation of the line integrals : $\int_{\gamma} f(x,y)\, dx$ $\int_{\gamma} f(x,y)\, dy$ (which should not be confused with $\int_{\gamma} f(x,y)\, ds$) where ...
0
votes
1answer
39 views

A question about improper integral

Could you please give me some hint how to solve this problem: Suppose f(x) continuous in $[0,\infty)$ and for each a,b>0 and c>b $ab \left|\int_0^1 f\left(ax+c \right) dx \right|<1$. Prove ...
8
votes
1answer
208 views

Cool Integral = $\pi/2$ !!

I am trying to calculate the integral $$ I_n=\int \limits_0^\infty \prod_{k=1}^n \frac{\sin \frac{x}{2k-1}}{\frac{x}{2k-1}}dx. $$ (I have literature on this, if people want). Note, we can write the ...
0
votes
1answer
19 views

Bartle - integration, monotone convergence theorem

Suppose that $(f_n) \subset M^{+}(X, \mathbb{X})$, that $(f_n)$ converges to $f$, and that $\int f d\mu=\lim \int f_n d\mu < +\infty$. Prove that $$\int_E f d\mu=\lim \int_E f_n d\mu $$ for each ...
0
votes
0answers
25 views

approximating a sphere

Suppose that $R$ is a simple connected region in $\mathbb{R}^3$, enclosing a volume $V$. I am looking at ways to approximate $V$ using spheroidal volume elements. The traditional approach is to use ...
1
vote
0answers
21 views

Use of mathematical induction to compute $\int_0^{\pi/2} \sin^{2n+1}xdx$?

We are given a reduction formula for $\int \sin^n xdx$, namely $$\int \sin^nxdx = -\frac{1}{n}\sin^{n-1}x\cos x + \frac{n-1}{n}\int \sin^{n-2}xdx.$$ I know how to derive this reduction formula. How ...
5
votes
1answer
49 views

Show that; $\displaystyle\int_0^\infty\frac{f(bx)-f(ax)}{x}dx=(l-f(0))\ln\frac{b}{a}$

Assume that $f\in C^1([0,\infty))$, $f'$ is monotonic on $[0,\infty)$ and $\lim\limits_{x\to\infty} f(x)=l$ is finite. If $a,b>0$, prove that; ...
7
votes
2answers
85 views

Is $\int_x^{\infty}e^{-\frac{t^2}{2}} < \frac{1}{x}e^{-\frac{x^2}{2}}$?

While solving a problem in real analysis, I got stuck. I need to prove $$\int_x^{\infty}e^{-\frac{t^2}{2}}dt < \frac{1}{x}e^{-\frac{x^2}{2}} $$ Clearly I have to use some kind of inequality, but ...
1
vote
1answer
22 views

Question about Fubini's Theorem for Riemann Integral functions

Let $f: [0,1] \times [0,1] \to \mathbb{R}$ be given such that $$ f(x,y) = \left\{ \begin{array}{lr} 1 & : x \in \mathbb{Q}\\ 2y & : x \notin \mathbb{Q} \end{array} ...
0
votes
0answers
26 views

rotation in the coordinate space x, y, z

Let $R = \{\ (x,y) : x \geq 1, 0 \leq y \leq \frac{1}{x} \}$. a) Prove that $R$ has finite area. b) Prove that when rotate $R$ in the coordinate space x, y, z about the x axis we get a solid of ...
4
votes
0answers
93 views

Beautiful Closed form $\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$

Hi I am trying to calculate $$ I:=\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$$ Note, the closed form is beautiful and is given by $$ I=−\frac{3}{8}\zeta_2\zeta_3 -\frac{2}{3}\zeta_2\ln^3 2 ...
2
votes
1answer
48 views

$\sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}$

Hi I am trying to calculate the sum given by $$ \sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}=\ = \sqrt{\frac{\pi}{\alpha}} e^{\beta^2/(4\alpha)} ...
5
votes
0answers
73 views

Beauty$\sum_{j,k=1}^\infty \frac{H_j(H_{k+1}-1)}{jk(k+1)(j+k)}=-\zeta(2)-2\zeta(3)+4\zeta(2)\zeta(3)+2\zeta(5)$

Hi I am trying to calculate the infinite double sum $$ S:=\sum_{j,k=1}^\infty \frac{H_j(H_{k+1}-1)}{jk(k+1)(j+k)}=-\zeta(2)-2\zeta(3)+4\zeta(2)\zeta(3)+2\zeta(5),\quad H_n:=\sum_{k=1}^n\frac{1}{k}\ \ ...
2
votes
2answers
52 views

how ro prove f(x,y) is integrable in $[a,b]\times[c,d]$

If there exits a $f(x,y)$ in $\mathbb{R}^2$,and if we fix any $x$ in $[a,b]$, then $f(x,y)$ is increasing as $y$ increases. Also, if we fix any $y$ in $[c,d]$,the $f(x,y)$ is increasing as $x$ ...
3
votes
0answers
35 views

Fundamental Theorem of Calculus and inverse..

If $F(x)$ is defined as $$F(x)= \int_{a}^{x} f(t) dt$$ calculate $(F^{-1})'(y)$ in terms of $f$. I have been working on this for a while now, does the aanswer to this incorporate the Inverse ...
1
vote
2answers
95 views

Question about Riemann Integration and the indicator function

Let $S \subseteq \mathbb{R}^n$. Suppose $\chi_S$ is integrable on $Q$ for some rectangle $Q$ such that $S \subseteq Q $. Let $\epsilon > 0 $ be given, I want to ask how can I show that there ...
0
votes
1answer
44 views

How to prove that $\max\{f,g\}$ is Riemann integrable? [duplicate]

If f(x) and g(x) are Riemann integrable in [a,b], why $h(x)=\max\{f(x),g(x)\}$ is still Riemann integrable in [a,b]? Or maybe it is wrong?
0
votes
1answer
31 views

a question about integral? I have no idea about that!

If f(x) and g(x) are integrable in [a,b], can we say that f(x)g(x) is still integrable in [a,b]? I am referring to Riemann integration!
1
vote
1answer
52 views

Suppose $f(x)\in L_1$ - Prove that $\lim_{n\rightarrow\infty}\int_0^\infty f(x)\cos(nx)dx = 0$

Assuming knowledge of the cyclic behavior of $cos(x)$, integration by parts, and $\int_0^{\infty} f<\infty$ is enough here? Consider \begin{align} & \int_0^\infty f(x)\cos(nx)dx = ...
2
votes
0answers
43 views

Log Cosine Integral $\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3)$

$$ I=\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3). $$ Note $\zeta(3)$ is given by $$ \zeta(3)=\sum_{n=1}^\infty \frac{1}{n^3}. $$ I have a ...
0
votes
0answers
22 views

Showing equivalence of weak convergence on closed and open intervals

Quick question. Let $I$ be an open bounded subset of $\mathbb{R}^{n}$. If I am given that $u_{m},u \in W^{1,\infty}(I)$ and I want to show that $u_{m} \rightharpoonup^{*} u$ in $L^{\infty}(I)$. Then I ...
1
vote
1answer
43 views

Compute $\int_{0}^x \vert \sin(t)\vert dt$ for $x\in \mathbb{R^+}$

Let $x\in \mathbb{R^+}$, compute $$\int_{0}^x \vert \sin(t)\vert dt$$ I tried like this : $$ \int_{0}^x \vert \sin(t)\vert dt=\int_0^{\lfloor \frac{x}{\pi}\rfloor \pi}\vert \sin(t)\vert ...
8
votes
0answers
220 views
+400

$\int_0^1 [ \frac{1}{x(x-1)} (2\mathrm{Li}_2(\frac{1-\sqrt{1-x}}{2})-\log(\frac{1+\sqrt{1-x}}{2})^2 ) -\frac{\zeta(2)-2\log^2 2}{x-1} ]dx$

Hi I am trying to evaluate $$ I:=\int \limits_{0}^{1} \left[ \frac{1}{x(x-1)} \bigg(2\mathrm{Li}_2\bigg(\frac{1-\sqrt{1-x}}{2}\bigg)-\log\bigg(\frac{1+\sqrt{1-x}}{2}\bigg)^2 \bigg) ...
1
vote
0answers
14 views

Prove with Lebesgue’s Criterion for integrablility that the composition $f\circ g$ is integrable

I have this homework question regarding Lebesgue's criterion for integrability and could use a bit of help. I'm not sure if my proof is entirely correct or formal enough. Here is said question: ...
7
votes
3answers
110 views

Showing that $\int_{0}^{\infty} \frac{dx}{1 + x^2} = 2 \int_0^1 \frac{dx}{1 + x^2}$

I was reading an article in which it was stated that, with a change of variable, one could show that: $$\int_{0}^{\infty} \frac{dx}{1 + x^2} = 2 \int_0^1 \frac{dx}{1 + x^2}$$ I tried with $t = 1 + ...
2
votes
1answer
21 views

What assumptions are needed to get two integrals close to each other?

I have functions $A,B,C$, where $\int_{\mathbb{R}} |A\cdot B - C| <\varepsilon$, and want to be able to say that $\int_{\mathbb{R}} A \approx \int_{\mathbb{R}} \frac{C}{B}$. What extra assumptions ...
1
vote
1answer
32 views

Let $S_n:= \frac{b-a}{n}\sum_{i=1}^{n}f(t_{i,n})$. Prove: $\lim_{n\to\infty}S_n = \int_a^bf(x)\ dx$.

I will post the assignment and then my attempt at solving it. Let $a,b \in \mathbb{R}$ with $a<b$ and let $f: [a,b] \rightarrow \mathbb{R}$ be a continous function. We'll now define a sequence ...
3
votes
3answers
72 views

Evaluate $\int \frac{\sqrt{x^2-1}}{x} \mathrm{d}x$

My try, using $x = \sec(u)$ substitution: $$ \begin{eqnarray} \int \frac{\sqrt{x^2-1}}{x} \mathrm{d}x &=& \int \frac{\sqrt{\sec^2(u) - 1}}{\sec(u)}\tan(u)\sec(u) \mathrm{d}u \\ &=& ...
-1
votes
1answer
67 views

Question concerning the integrability of a function

Let $f: [0,1]^2 \to \mathbb{R}$ be a function such that $$ f(x,y) = \left\{ \begin{array}{lr} 1 & : x \in \mathbb{Q} \\ 2y & : x \notin \mathbb{Q} \end{array} ...
2
votes
1answer
52 views

Integral $\int_0^{\pi/2} \log^n (\sin t)\log^p (\cos t) dt$

I am looking for a closed form expression for the logarithmic trigonometric integral $$ I_{n,p}=\int_0^{\pi/2} \log^n (\sin t)\log^p (\cos t) dt \quad (n\geq 0, p\geq 0). $$ Closed form expression ...
10
votes
2answers
163 views

Calculate $\int_0^1 \frac{\ln(1-x+x^2)}{x-x^2}dx$

I am trying to calculate: $$\int_0^1 \frac{\ln(1-x+x^2)}{x-x^2}dx$$ I am not looking for an answer but simply a nudge in the right direction. A stradegy, just something that would get me started. ...
1
vote
2answers
139 views

Real analysis question involving inhomogenous linear ODE

So I had another problem like this but the ODE was homogenous, now there is a non zero right side. I completed part (i), $\large c(x) = \int \frac{b(x)}{g(x)} dx$. I am stuck on (v). (1) is the ...
1
vote
1answer
42 views

Prove by using step functions: $\int_{-b}^{b}\sin(x)\ dx = 0$

The Assignment: Let $b > 0$. Prove by using step functions: $$\int_{-b}^{b}\sin(x)\ dx = 0$$ The claim itself is obvious, but I have no idea how to prove it with step functions. My idea was ...
1
vote
1answer
49 views

integral $I=\int_{-\infty}^\infty e^{-\alpha x^{2k}}dx$

$$ I=\int_{-\infty}^\infty e^{-\alpha x^{2k}} dx $$ The last problem was ill posed, and is answered in the post! You can disregard this post!
9
votes
0answers
205 views
+400

$I=\frac{1}{\pi}\int_0^{\pi/3}\log\big( \mu(\theta)+\sqrt{\mu^2(\theta)-1} \big)\ d\theta, \quad \mu(\theta)=\frac{1+2\cos\theta}{2}.$

Hi I am trying to calculate this integral: $$ I=\frac{1}{\pi}\int_0^{\pi/3}\log\left( \frac{1+2\cos\theta}{2}+\sqrt{\bigg( \frac{1+2\cos\theta}{2} \bigg)^2-1} \right)\ d\theta. $$ The ...
0
votes
0answers
21 views

Integration over rectangles

"Prove that a function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is integrable over every rectangle in $\mathbb{R}^n $ if and only if it is integrable over every ball in $\mathbb{R}^n$" So I'm stumped ...
0
votes
1answer
46 views

Double Integral Proof

Let function $f(x, y)$ be defined by $$f(x, y) =\begin{cases} 1,\text{ if }x = y,\\ 0,\text{ otherwise}.\end{cases}$$ Using the definition of the double integral show that the following integral exists ...