Questions about the evaluation of specific definite integrals.

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2
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1answer
32 views

Show that $ \int_0^2 e^{x^2-x} dx \in [2e^{-1/4},2e^2] $

Show that $ \int_0^2 e^{x^2-x} dx \in [2e^{-1/4},2e^2] $ If $f(x)\leq g(x)$ for $x\in[a,b]$ then $\int^b_af(x)dx\leq \int^b_ag(x)dx$ if $x\in [0,2]$ then $x^2-x\leq x$, so $$0 \leq \int_0^2 ...
7
votes
2answers
121 views

A reason for the value of $\int_{0}^{1}\log{(x)}\log{(1-x)}\,\mathrm{d}x$

In this .pdf document, which is just a list of Putnam-style undergraduate-level problems from various sources, the third question is as I have stated it below (up to a change of notation). ...
2
votes
0answers
14 views

Definite integral with modified Bessel functions of first and second kind

I am interested in the following integral involving the modified Bessel functions of the first and second kinds of order one $I = \int_0^{\infty} \frac{\sin(ax)}{x} I_1(bx) K_1(cx) \mathrm{d}x$ For ...
1
vote
2answers
122 views

Solve this integral:$\int_0^\infty\dfrac{\arctan x}{x(x^2+1)}\mathrm dx$

I occasionally found that $\displaystyle\int_0^{\frac{\pi}{2}}\dfrac{x}{\tan x}=\dfrac{\pi}{2}\ln 2$. I tried that $$\int_0^{\frac{\pi}{2}}\dfrac{x}{\tan x}=\int_0^{\frac{\pi}{2}}x \ \mathrm ...
0
votes
0answers
32 views

A little hard double integral

$\iint \frac{2x^2e^{x^2}}{x^2+y^2}dxdy\::\:D=\left\{1\le x\le 2,\:0\le y\le x\right\}$ I use the substitution: $u=x^2,\:v=\frac{y}{x}$ $$$$Then I get: ...
1
vote
1answer
39 views

Calculating $\iint_{D} \left(x-y\right)dxdy$ where $D=\left\{0\le x-y\le 1,\:1\le xy\le 2\right\}$

$$\iint_{D} \left(x-y\right)dxdy$$ where $D=\left\{0\le x-y\le 1,\:1\le xy\le 2\right\}$ So the substitution is pretty obvious, but j is: $J\:=\frac{1}{x+y}$ $$$$ I dont see how I get rid of the ...
1
vote
1answer
58 views

If $f$ is differentiable and $f'$ is bounded then relation between upper sum , lower sum and the integral

Let , $f:\mathbb R\to \mathbb R$ be a differentiable function such that $f'$ is bounded. Given a closed and bounded interval $[a,b]$ and partition $P=\{a=a_0<a_1<\cdots <a_n=b\}$ of $[a,b]$ . ...
4
votes
0answers
79 views

Another integral related to Fresnel integrals

How would we prove this result by real methods ? $$\int_0^{\infty } \frac{\sin \left(\pi x^2\right)}{x+2} \, dx=\frac{1}{4} \left(\pi-2 \pi C\left(2 \sqrt{2}\right)-2 \pi S\left(2 ...
1
vote
0answers
29 views

Differentiating the definite integral of x*f(x)

I'm trying to differentiate the integral below. I was wondering how I could approach it. z ~ N(a , $b^2$ $\cdot$ $x^{-2}$) $\frac{d}{dx} \int _{-\infty} ^ {z^*(x)} z \cdot \Phi ^{\prime} (z) dz$ ...
2
votes
2answers
109 views

How to solve $ \int \limits _0 ^{2\pi} \dfrac{dx}{(\alpha +\beta\cos x)^2} $

I am trying to solve this integral, I think that it could be solve using the complex. $$ \int \limits _0 ^{2\pi} \dfrac{dx}{(\alpha +\beta\cos x)^2} $$
0
votes
3answers
28 views

Find the area between the given function , and two tangents off of the point (2,-2)

So here is a general graph of the first couple directions. $T_1$ and $T_2$ are supposed to be the points where the tangent line intersects the parabola. The tangent lines and points where the ...
3
votes
1answer
99 views

Calculating $\int_0^{\pi/4} \frac{\cot (x)}{\cot ^2(x)+\sqrt{\cot (x)}} \, dx$

This is not really one of that kind of integrals that Mathematica cannot handle with, but given the case of a contest, how would we like to handle with it? I would like so much to know your ideas ...
1
vote
0answers
36 views

Integrals with error function and exponentials

I'm trying to solve the integrals below: $$\int_{-\infty}^\infty \int_{-\infty}^\infty \frac{x}{\sqrt{x^2+y^2}}\cdot \operatorname{erf}\left(m\cdot\sqrt{x^2+y^2}\right) \cdot \exp(-a\cdot ...
4
votes
2answers
77 views

Evaluating definite integral of $e^{i t^2}$

In passing Sakurai's QM book mentions that $$\int_{-\infty}^\infty e^{i t^2} dt = \sqrt{i \pi}$$ This is consistent with 7.4.4 in Abramowitz and Stegun which claims for $\Re a > 0, n = 0, 1, 2, ...
0
votes
0answers
29 views

Solving for the limit of a Gaussian random variable within an integral

I'm having trouble solving a particular integral. It is $$ (1/\Delta t)\int_t^{t+\Delta t}I(t')dt', $$ where $$ I(t') = \mu_c+\sigma_c \eta(t'). $$ In this second equation, $$ \eta(t') = ...
4
votes
0answers
81 views

Evaluating $\int_0^{\pi /2}\left(\frac{1}{\sqrt{\tan(x)}}+\frac{1}{\sqrt{\arctan(x)}}\right) dx$ [on hold]

I've come across the following integral: $$\int_0^{\pi /2}\left(\frac{1}{\sqrt{\tan(x)}}+\frac{1}{\sqrt{\arctan(x)}}\right) dx$$ I haven't been able to make any of the obvious methods work (or make ...
3
votes
1answer
37 views

bounding a sum using a definite integral

Conjecture. Let $1<p<\infty$. Then there exists $C\in(0,\infty)$ such that for any $k\in\mathbb{Z}^+$ we have \begin{equation}\sum_{n=1}^k(k+1-n)^{-\frac{p}{p+1}}n^{-\frac{1}{p+1}}\leq ...
10
votes
3answers
162 views

Calculating $\int_0^{\infty } \left(\text{Li}_2\left(-\frac{1}{x^2}\right)\right)^2 \, dx$

Do you see any fast way of calculating this one? $$\int_0^{\infty } \left(\text{Li}_2\left(-\frac{1}{x^2}\right)\right)^2 \, dx$$ Numerically, it's about $$\approx ...
13
votes
3answers
159 views

Prove that $\int_0^1 \frac{1}{1+\ln^2 x}\,dx = \int_1^\infty \frac{\sin(x-1)}{x}\,dx $

I've found the following identity. $$\int_0^1 \frac{1}{1+\ln^2 x}\,dx = \int_1^\infty \frac{\sin(x-1)}{x}\,dx $$ I could verify it by using CAS, and calculate the integrals in term of ...
1
vote
1answer
23 views

Find the values of the derivatives of the integral with a variable inside its limits.

$\require{cancel}$ Problem: I have the function $g: \mathbb{R} \to \mathbb{R}$ defined as $$ g(x)=\int^{(1+x^2)}_{-(1+x^2)} sin(t^3)\ dt,\ x \in \mathbb{R} $$ I would like to calculate values of ...
2
votes
1answer
45 views

Convergence of an integration $t=\int_{x_0}^{x_1}\sqrt{\frac{m}{2(E_0-V(y))}}dy$

When I am reading Brian Hall's "Quantum Theory for Mathematicians", I came across an integration (frequently appeared in physics textbooks) $$t=\int_{x_0}^{x_1}\sqrt{\frac{m}{2(E_0-V(y))}}dy.$$ The ...
13
votes
4answers
229 views

A conjectured result for $\sum_{n=1}^\infty\frac{(-1)^n\,H_{n/5}}n$

Let $H_q$ denote harmonic numbers (generalized to a non-integer index $q$): $$H_q=\sum_{k=1}^\infty\left(\frac1k-\frac1{k+q}\right)=\int_0^1\frac{1-x^q}{1-x}dx=\gamma+\psi(q+1),\tag1$$ where ...
0
votes
1answer
28 views

Transform integral limits

Assume that the following relation holds \begin{align} p\int^1_{\frac{p}{a}} f(\theta)d\theta = \int^p_a f(\theta)d\theta \end{align} where $p$ is some scalar and $a\in[0,1)$. Is there some general ...
2
votes
5answers
103 views

definite integral of $x^2e^{-x^2}$

I am trying to calculate the integral of this form: $\int_{-\infty}^{+\infty}e^{-x^2}\cdot x^2dx$ I am stuck. I know the result, but I'd like to know the solution step-by-step, because, as some ...
2
votes
2answers
22 views

Calculating values of integrals using Fourier series and uniform convergence

I have a problem that I don't know how to begin solving. I have f(t) $$ f(t) = \sum_{k=1}^\infty\frac{1}{k^2+1}\sin{kt} $$ First I had to show that this series converges uniformly, I've done that ...
3
votes
2answers
140 views

Proving that $\int_0^{\pi/2} (\sin (\tan (x))+\cot (x) \cos (\tan (x))-\cot (x))\cot (x) \, dx=\frac{\pi(e-2)}{2e}$

I think one of the ways of doing it is by the use of the differentiation with parameter. Do you see an easy way of calculating it by real methods? $$\int_0^{\pi/2} (\sin (\tan (x))+\cot (x) \cos ...
0
votes
1answer
11 views

Let $R$ be the region in the first quadrant bounded by the $x$ and $y$ axis and the graphs of $f(x)=\frac{9}{25}x+b$ and $y=f^{-1}(x)$

Let $R$ be the region in the first quadrant bounded by the $x$ and $y$ axis and the graphs of $f(x)=\frac{9}{25}x+b$ and $y=f^{-1}(x)$.If the area of $R$ is 49,then the value of $b$,is ...
1
vote
1answer
52 views

Integration of a analytic function

here is the problem I currently try to solve: $$\int\limits_{-\infty}^{+\infty}\left((1+ixa^2)^{-\frac{n_1}{2}}\cdot(1+ixb^2)^{-\frac{n_2}{2}}\right)e^{icx} \mathrm{d}x $$ with $a,b,c\geq0$ (real ...
0
votes
0answers
12 views

Integral of the product of two Theta functions

Is there a closed form expression for the integral $\int_{-\infty}^{\infty}dx \Theta(k^2-|x|)\Theta(k^2-|x+a+b\sqrt{x+1}|)$ or $\int_{-k^2}^{k^2}dx \Theta(k^2-|x+a+b\sqrt{x+1}|)$ where ...
4
votes
0answers
63 views

On finding an explicit form of a particular recurrence relation

Let $f$ be integrable over the interval $[0, 1]$, and $$I_n = \int_0^{1} x^n f(x) \, \mathrm{d}x.$$ Suppose $f(x) = f(1-x)$; we can then show that $$I_n = \sum_{k=0}^{n} \binom{n}{k} (-1)^k \, ...
0
votes
3answers
57 views

Why is this integral “inconsistent”?

I am currently working on the following integral: $$\int_{-\infty}^{\infty} xe^{-|x+1|} dx$$ The method I use for it is Integration by Parts. When I calculate the integral $\int_{-\infty}^{\infty} ...
1
vote
2answers
73 views

Conceptual differences between the notations $\int_{a}^{b}f$ and $\int_{[a,b]}f$

Let $[a,b] \subset \mathbb{R}$ and let $f: [a,b] \to \mathbb{R}$ be continuous. Then $f$ is Riemann-integrable. What are the conceptual differences between the two notations $\int_{a}^{b}f$ and ...
0
votes
1answer
28 views

Choosing a technique for solids of revolution

Is there a heuristic to choose between the disk method and the washer method? To take the simplest example $y=x$ can be revolved around the $y$ axis using $R=x$ or $r=x$ and $R=$ a constant $C$.
4
votes
2answers
89 views

Double integral with a product of dilog $\int _0^1\int _0^1\text{Li}_2(x y) \text{Li}_2((1-y) x)\ dx \ dy$

One of the integrals I came across these days (during my studies) is $$\int _0^1\int _0^1\text{Li}_2(x y) \text{Li}_2((1-y) x) \ dx \ dy$$ that can be turned into a series, or can be approached by ...
1
vote
1answer
58 views

Correctness of the definite integral

Consider the integral \begin{eqnarray*} I & = & \int_{-1}^{1}\frac{dx}{\sqrt{1-x^{2}}(1+\sqrt{1-x^{2}})}\\ & = & ...
4
votes
1answer
61 views

closed-form of an integral similar to Bessel function

The integral form of the $n$-th modified Bessel function of the first kind is $$ I_n(z)=\frac{1}{\pi}\int_0^{\pi}e^{z\cos\theta}\cos(n\theta)\;d\theta. $$ However, I found an integral $$ ...
2
votes
1answer
40 views

Estimate integral $\,\displaystyle\int_{0}^{\infty}\operatorname{sech}\left(\varepsilon x\right)\cos\left(kx\right)\,dx,\,$ with $\,k,\varepsilon>0$

$ \newcommand{\sech}{\operatorname{sech}} $ Is there any analytic/asymptotic way to estimate the value of the integral: $$ \int_{0}^{\infty} \sech\left(\varepsilon x\right)\cdot ...
2
votes
2answers
32 views

Let $ f$ be a function such that $|f(u)-f(v)|\leq|u-v|$ for all real $u$ and $v$ in an interval $[a,b]$

Let $ f$ be a function such that $|f(u)-f(v)|\leq|u-v|$ for all real $u$ and $v$ in an interval $[a,b]$.Then: $(i)$Prove that $f$ is continuous at each point of $[a,b]$. $(ii)$Assume that $f$ is ...
2
votes
1answer
87 views

$\int_{0}^{1}\frac{x^{n-1}}{1+x^n} \log\log{(\frac{1}{x})}dx =-\frac{\log(2)\log(2n^2)}{2n}$ [on hold]

Does anyone prove the following definite integral ? $$\int_{0}^{1}\frac{x^{n-1}}{1+x^n} \log\log{(\frac{1}{x})}dx =-\frac{\log2(\log2+2\log(n))}{2n}$$
1
vote
2answers
52 views

show that $F''$ is strictly increasing.

If $f$ is continuous and always positive in $[0,\infty)$ and $$F(x)=\frac{1}{2}\int_0^x(x-t)^2f(t)\,dt$$then show that $F''$ is strictly increasing. I found that the integrand is continuous and ...
2
votes
2answers
65 views

Given that $ U_n=[x(1-x)]^n$ and $n\geq2$,$V_n=\int_{0}^{1}e^xU_ndx$,prove that $V_n+2n(2n-1)V_{n-1}-n(n-1)V_{n-2}=0$

Given that $ U_n=[x(1-x)]^n$ and $n\geq2$,$V_n=\int_{0}^{1}e^xU_ndx$,prove that $V_n+2n(2n-1)V_{n-1}-n(n-1)V_{n-2}=0$ I tried to solve it by integration by parts,taking $U_n$ as first function and ...
2
votes
3answers
36 views

Prove that $(1)\frac{1}{2}\leq\int_{0}^{2}\frac{dx}{2+x^2}\leq\frac{5}{6}$<br> $(2)2e^{-1/4}<\int_{0}^{2}e^{x^2-x}dx<2e^2$

Prove that $(1)\frac{1}{2}\leq\int_{0}^{2}\frac{dx}{2+x^2}\leq\frac{5}{6}$ $(2)2e^{-1/4}<\int_{0}^{2}e^{x^2-x}dx<2e^2$ I tried to prove it but my answer is not correct. For first part,As ...
1
vote
3answers
41 views

Define g(x) as a function of $x$

Let $f(x)= \begin{cases} -1 & ,-2\leq x\leq 0 \\ \\ |x-1| & ,0<x\leq2 \end{cases}$ and $g(x)=\int_{-2}^{x}f(t) dt$.Define g(x) as a function of $x$. I tried to ...
0
votes
1answer
22 views

Discuss the following relation s about upper and lower Riemann sums

Let , $f$ be a continuously differentiable real-valued function on $[a,b]$ such that $|f'(x)|\le k $ for all $x\in [a,b]$. For a partition $P=\{a=a_0<a_1<a_2<\cdots <a_n=b\}$ let ...
1
vote
2answers
27 views

Relations between upper and lower Riemann sum

Let , $f$ , $g$ , $h$ be bounded functions on the closed interval $[a,b]$ such that $f(x)\le g(x)\le h(x)$ for all $x\in [a,b]$. Let , $P=\{a=a_0<a_1<a_2<\cdots <a_n=b\}$ be a partition ...
6
votes
2answers
185 views
2
votes
0answers
38 views

Please help me understand Analytic Density $\lim_{\sigma \to 1^+}\frac{1}{\zeta(\sigma)}\sum_{n \in A} \frac{1}{n^{\sigma}}$

$d (A) = \lim_{\sigma \to 1^+}\frac{1}{\zeta(\sigma)}\sum_{n \in B} \frac{1}{n^{\sigma}}$ for $B \subset \Bbb{N}$. So clearly this limit is $0$ for reciprocally summable (convergent) $B$. My goal ...
1
vote
2answers
35 views

Find the range of the function,$f(x)=\int_{-1}^{1}\frac{\sin x}{1-2t\cos x+t^2}dt$

Find the range of the function,$f(x)=\int_{-1}^{1}\frac{\sin x}{1-2t\cos x+t^2}dt$ I tried to solve it,i got range $\frac{\pi}{2}$ but the answer is ${\frac{-\pi}{2},\frac{\pi}{2}}$ ...
4
votes
5answers
59 views

Determine a positive integer $n\leq5$,such that $\int_{0}^{1}e^x(x-1)^ndx=16-6e$

Determine a positive integer $n\leq5$,such that $\int_{0}^{1}e^x(x-1)^ndx=16-6e$. I tried to solve it.But since $n$ is given to be $\leq$ 5,my calculations went lengthy. Applying integration by ...
2
votes
2answers
39 views

One integral involving integrals exponential and logarithmic function

Is there a closed-form solution for the integral $$ \int_{0}^{\infty}\log_{2}(1+ax)\cdot e^{-bx} \; \mathrm dx $$ with $a, b \geq 0$? If there is no closed-form solution, whether there is an ...