Questions about the evaluation of specific definite integrals.

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2
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1answer
15 views

Integral of Bessel Functions Multiplying “polynomials”

How can I compute the following integral: $$\int_{0}^{1} (1 - x^{2})^{\nu - \mu - 1} x^{\mu + 1} J_{\mu}(\alpha_{\nu}x) dx$$ where $\nu > \mu \geq 1$ and $J_{\nu }(\alpha_{\nu}) = 0$. The $J_{\nu}$ ...
2
votes
3answers
91 views

How do I prove $\int_{-\infty}^{\infty}{\cos(x+a)\over (x+b)^2+1}dx={\pi\over e}{\cos(a-b)}$?

How do I prove these? $$\int_{-\infty}^{\infty}{\sin(x+a)\over (x+b)^2+1}dx={\pi\over e}\color{blue}{\sin(a-b)}\tag1$$ $$\int_{-\infty}^{\infty}{\cos(x+a)\over (x+b)^2+1}dx={\pi\over e}\color{blue}{\...
1
vote
1answer
24 views

Calculating $\sum_{n=1}^x\frac{r^n}{n^k}$ with integrals

Through some work, I've managed to solve the following sum in the form of integrals: $$\sum_{n=1}^x\frac{r^n}{n^k}=\int_0^r\frac1{a_{k-1}}\int_0^{a_{k-1}}\frac1{a_{k-2}}\int_0^{a_{k-2}}\dots\int_0^{...
2
votes
1answer
40 views

Need help with an integral involving the Dirac delta function

I'm having trouble evaluating this integral, which involves the Dirac delta function: $$ \int\limits_{0}^\infty \frac{\cos(\pi x)}{x} \delta \left[ (x^2-1)(x-2) \right] \mathrm{d}x $$ I think I ...
0
votes
1answer
43 views

How to compute the definite integral $\int _0^{\infty }\:\frac{\left(2e^x+1\right)}{e^{2x}+2e^x+2}dx $

Good evening to everyone. I have an integral that I don't know how to compute: $$ \int _0^{\infty }\:\frac{\left(2e^x+1\right)}{e^{2x}+2e^x+2}dx $$.I've never computed integrals with $ \infty$ before ...
0
votes
1answer
80 views

Value of the integral $\int_{0}^{1}x^{n}(1-x)^{n}dx$

If $A=\int_{0}^{1}x^{n}(1-x)^{n}dx$ then which of the following is/are true? $1.$ $A$ is not a rational number. $2. 0<A\leq 4^{-n}.$ $3.$ A is a natural number. $4.$ $A^{-1}$ is a natural ...
8
votes
1answer
116 views

Daunting series of integrals: $\sum_{n=2}^\infty\int_0^{\pi/2}\sqrt{\frac{(1-\sin x)^{n-2}}{(1+\sin x)^{n+2}}}\log(\frac{1-\sin x}{1+\sin x})dx$

My coleague showed me the following integral yesterday \begin{equation} I=\sum_{n=2}^{\infty}\int_0^{\pi/2}\sqrt{\frac{(1-\sin x)^{n-2}}{(1+\sin x)^{n+2}}}\log\left(\!\frac{1-\sin x}{1+\sin x}\!\...
0
votes
1answer
50 views

Compute definite integral by hand [on hold]

How can I compute $$\int_0^1 \frac{x^3t}{(x^2+t^2)^2} \, \mathrm{dt}$$ by hand?
0
votes
0answers
11 views

Solving the definite integral $\int_{-\psi/2}^{\psi/2}\,\exp{(A\sin(\theta+\phi))}d\theta$

I need to solve this definite integral: $$\int_{-\psi/2}^{\psi/2}\,\exp{(A\sin(\theta+\phi))}d\theta$$ where $A$ is a real positive constant and $\psi\in[0,2\pi]$. I know that for $\psi=2\pi$ the ...
8
votes
3answers
164 views

How to prove that$\int_{0}^{1}\ln{(x/(1-x))}\ln{(1+x-x^2)}\frac{dx}{x}=-\frac{2}{5}\zeta{(3)}$

$$\int_{0}^{1}\ln{\big(\frac{x}{1-x}\big)}\ln{(1+x-x^2)}\frac{dx}{x}=-\frac{2}{5}\zeta{(3)}$$ Put $$\frac{x}{1-x}=y$$ $$I=\int_{0}^{\infty}\ln{y}\ln{(1+3y+y^2)}\frac{dy}{y(y+1)}=\frac{8}{5}\zeta{(3)...
2
votes
3answers
54 views

Showing that $\int_{-n}^{n}{x+\tan{x}\over A +B(x+\tan{x})^{2n}}dx=0$

Where n is an integer, $n\ge1$ and $(A,B)$ just constants $$I=\int_{-n}^{n}{x+\tan{x}\over A +B(x+\tan{x})^{2n}}dx=0$$ It is obvious that $$\int_{-n}^{n}x+\tan{x}dx=0$$ Let make a ...
2
votes
0answers
18 views

Second Mean Value Theorem for Integrals, Variant

I want to prove the following theorem, which Wikipedia refers as 'Second Mean Value Theorem' Suppose that $g(x)$ is a non-negative monotonically decreasing function on the interval $[a, b]$, and ...
2
votes
1answer
16 views

Deriving the mean of the Gumbel Distribution

I'm trying to determine an expected value of a random variable related to the Gumbel/Extreme Value Type 1 distribution. I think the answer follows the same process as expected value of the Gumbel ...
5
votes
1answer
96 views

finding the series $\sum_{n=1}^\infty \frac{x^n}{n!} \frac{1}{n}$

My goal is to solve this series $$S(x) = \sum_{n=1}^\infty \frac{x^n}{n!} \frac{1}{n}$$ I did took the derivative first w.r.t $x$ $$S'(x) = \sum_{n=1}^\infty \frac{x^{n-1}}{n!}$$ which I ...
0
votes
3answers
53 views

Proving a Definite Integral Inequality without Geometrical Intuition

I solved an integral inequality problem using geometrical methods. However, I just cannot satisfy with them and want a without-geometrical-intuition proof, and I couldn't find one. Proof the ...
0
votes
1answer
44 views

Calculation of double integral

I am trying to solve this integral $$ \int_{20}^{21}\int_{20}^{25}\frac{1}{\sqrt{2\pi}ga_{m}}\exp\Big{(}-\frac{1}{2} \frac{(a_{m}-(ba_{f}+c))^{2}}{g^{2}a_m^{2}}\Big{)}da_{m}da_{f} $$ with b,c,g ...
0
votes
0answers
22 views

Evaluate an integral with hyperbolic functions

I am trying to evaluate, respectively simplify the following integral expression $$f(t):=\int_{a}^{t-a} \frac{(\cosh(x-a)-\cosh(a))^{i\tau-1/2}}{(\cosh(x)-\cosh(a) )^{i\tau+1/2}} dx, \quad t>a,$$ ...
0
votes
1answer
51 views

How does one compute this heavy integral?

The integral is $$\frac{1}{2\pi i}\int_\Gamma\frac{\exp(z^2-\cos(iz)-4)}{z-2}dz$$ where $\Gamma$ is the unit circle. Here's how I tried to parametrize it: $z=e^{i\theta}$ on $\theta\in [0, 2\pi]$, ...
3
votes
3answers
82 views

Volumes of Revolutions : Lord of the Rings

Question: The "Lord of the Rings" has a collection of solid gold rings for different-sizes fingers. The cross section of each ring is a segment of a circle radius $R$ as shown in the diagram below. ...
1
vote
0answers
43 views

Double Gaussian Integral

I am interested in the following integral $$\int_{-a}^adx\int_{-a}^a\mathop{\mathrm{d}y}xy\frac{1}{\sqrt{b}}\exp\left[-\frac{(x-y)^2}{2b}\right]\sqrt{\frac{2}{c}}\exp\left[-\frac{(x+y)^2}{4c}\right].$...
5
votes
1answer
114 views

How do I find a closed form of ${\pi^{2n}\over \zeta(2n)}\int_{-1}^{1}{x^{2n-2}\over \pi^2+(2\tanh^{-1}{x})^2}dx$?

How do I evaluate the closed form for $g(n)$? Where n is an integer, $n\ge 1$ $${\pi^{2n}\over \zeta(2n)}\int_{-1}^{1}{x^{2n-2}\over \pi^2+(2\tanh^{-1}{x})^2}dx=g(n)$$ Make a subsititution $u=\...
1
vote
0answers
39 views

difficult integral with bessel functions

$$\int_{0}^R \rho J_0\left(\frac \rho {\sqrt \tau} \right) J_0 \left(k_n \rho \right) d\rho = -\frac {{R^3}/{\sqrt \tau}}{x_n^2-{R^2}/{\sqrt \tau}} J_1\left(\frac{R}{\sqrt \tau}\right) J_0(x_n)$$ I ...
3
votes
2answers
53 views

limit of an integral over a function

Calculate $\displaystyle\lim_{x\to0}{F(x)\over g(x)}$, where $ g(x)=x$ and $\displaystyle F(x)=\int_0^x {e^{2t}-2e^t+1\over 2\cos3t-2\cos2t+\cos t} \, dt$. i'd love for someone to explain not only ...
2
votes
1answer
57 views

Doing $\int_V^{\alpha} \frac{dE}{\sqrt{(\alpha - E)(E - V)}} = \pi$ without Work

Is there a way to see that $$\int_V^{\alpha} \frac{dE}{\sqrt{(\alpha - E)(E - V)}} = \pi$$ should be true without doing any work? Is there a quick way to brute-force this without long computations? ...
3
votes
1answer
62 views

How to evaluate this Fourier Transform $A\int_{-\infty}^{\infty} \frac{e^{ikx}}{(1+x^{2})^{\frac{\nu+1}{2}}}dx$

This is basically the Fourier transform of a Student´s T pdf. How do we compute it? $$A\int_{-\infty}^{\infty} \frac{e^{ikx}}{(1+x^{2})^{\frac{\nu+1}{2}}}dx$$ for $\nu$ any number greater than zero ...
2
votes
0answers
33 views
+150

Integral involving the von Mises-Fisher distribution

I'm going quickly through the VonMises-Fisher distribution $M$ on $\mathbb S^{d-1}$ and its properties. Its probability density function is: $$f(x; \kappa,\mu)= c(\kappa)\exp(\kappa x^T\mu)$$ where $...
13
votes
5answers
183 views

I want to show that $\int_{-\infty}^{\infty}{\left(x^2-x+\pi\over x^4-x^2+1\right)^2}dx=\pi+\pi^2+\pi^3$

I want to show that $$\int_{-\infty}^{\infty}{\left(x^2-x+\pi\over x^4-x^2+1\right)^2}dx=\pi+\pi^2+\pi^3$$ Expand $(x^4-x+\pi)^2=x^4-2x^3+2x^2-2x\pi+\pi{x^2}+\pi^2$ Let see (substitution of $y=x^2$)...
0
votes
1answer
19 views

Alternate Definition of Definite Integral

I was trying to solve this given problem, When $f(x)$ is continuous on $[a, b]$, there exists infinitely many reals $p_1, p_2, p_3$ and $q_1, q_2, q_3, q_4$, which satisfies the following ...
0
votes
1answer
56 views

Riemann's sum with natural logarithm

I have this problem - Express limit as a definite integral. The limit is: $$\lim_{n\to \infty}\sum_{i=0}^n\frac{3+\ln(n+i)-\ln(n)}{n}$$ I can't figure out how to make a definite integral out of it. ...
1
vote
1answer
20 views

Interpreting the unit of definite integral

I'm calculating a definite integral of a function describing a pressure of certain system where time is the independent variable, that is my function is: $$f(t) \;\text{psi},$$ so e.g. $f(0)\;\text{...
1
vote
2answers
32 views

evaluation of this exponential integral?

how could i evaluate $$ \int_{0}^{a(E)}\sqrt{E-16\pi ^{2}e^{4x}} $$ where 'a' is the point so $ E-16\pi^{2}e^{4a}=0 $ this appear s in Quantum mechanic so i think the answer is something like $$ E^...
0
votes
2answers
68 views

Integrate logarithmic derivative of a periodic function

Given $f$ a $p$-periodic function over $\mathbb{C}$, how to show that : $$\frac{1}{\mathrm{i}p}\int_a^{a+p}\frac{f'(t)}{f(t)}dt \in \mathbb{Z}$$ Is there any elegant method ?
1
vote
6answers
178 views

Solve definite integral: $\int_{-1}^{1}\arctan(\sqrt{x+2})\ dx$

I need to solve: $$\int_{-1}^{1}\arctan(\sqrt{x+2})\ dx$$ Here is my steps, first of all consider just the indefinite integral: $$\int \arctan(\sqrt{x+2})dx = \int \arctan(\sqrt{x+2}) \cdot 1\ dx$$ ...
1
vote
1answer
30 views

Volume by integration - Disk Method only/Non-coordinate axis

PROBLEM: Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line x = 5. (Use disk method) $$ xy = 3, y = 1, y = 4, x = 5 $$ So first I ...
1
vote
1answer
46 views

how to get the function(s) under the integral sign in definite integral

Say we have the definite integral: $$\int_a^b{f(x)\, \mathrm{d}x} = \alpha$$ Given, $a, b,$ and $\alpha \in \mathbb{R}$, is it possible to get the functions $f(x)$ in general case? Thank you
1
vote
0answers
51 views

Find the Log and dfrac integral closed form? [on hold]

Find the $$f(a)=\int_{0}^{+\infty}\dfrac{\ln^2{\left(\cot{\left(\dfrac{ax}{2}-\frac{\pi}{4}\right)}\right)}}{x^4+4}dx$$ where $a>0$ Try this wwo classic methods $x\to\dfrac{1}{x}?$ or $f'(a)$ ...
4
votes
3answers
112 views

Find all continuous functions $f:[0,1]\rightarrow \mathbb{R}$ that satisfy: $\int_0^1 f(x)dx=1/3 + \int_0^1 f^2(x^2)dx$

(Note that $f^2(x)=f(x)\cdot f(x)$ and not composition.) Since both integrals are defined, derivation is out of the question. I tried integrating the second integral by parts but reached something ...
1
vote
2answers
53 views
+50

Double Dirac delta integral

Let $I=\int_{-1}^0\int_{0}^1 \delta(x-y)dxdy$ where $\delta(t)$ is defined as the limit of a symmetric Gaussian pdf. The ranges overlap only on the zero-length range $x=y=0$. Is the result here $0$ ...
0
votes
4answers
105 views

Value of $\int\tan^{-1}(x)\,dx$

What is the value of $\int^{1000}_{0}\tan^{-1}(x)\,\mathrm d x$? Today we were taught about graphs of all trigonometric inverse functions. So my proofessor split it into $0-\tan(1)$ and $\tan(1)-...
0
votes
4answers
120 views

Strange integral result

Consider the following integral, $$\mathrm{I} = \int_{-1}^{1}\frac{d}{dx}\tan^{-1}\left(\frac{1}{x}\right)dx$$ We can do this in two ways, First Using the fact that the antiderivative of $\frac{d}{...
1
vote
0answers
55 views

How can I evaluate the definite integral [duplicate]

Evaluate the definite integral: $$\sqrt{\frac{2}{\pi}} \int^{\infty}_{t} \frac{x^2}{x^2+1} e^{-\frac{x^2}{2}}dx$$ The form is a little different from the existing question How to Solve this definite ...
2
votes
0answers
70 views

How to Solve this definite integral

I have a problem to evaluate this definite integral. Evaluate the definite integral: $$\sqrt{\frac{2}{\pi}} \int^{\infty}_{t} \frac{1}{x^2+1} e^{-\frac{x^2}{2}}dx$$
1
vote
3answers
68 views

Average value of $f(x)=\int_x^1 \cos(t^2) dt$ on the interval $[0,1]$.

Find the average value of the function $$f(x)=\int_x^1 \cos(t^2) dt $$ on the interval $[0,1]$.
1
vote
3answers
132 views

How to integrate $\int _1^{\infty }\frac{dx}{\left(x^2+1\right)\sqrt{x^2-1}}= \;?$

How do I integrate $\int _1^{\infty }\left(\frac{1}{\left(x^2+1\right)\sqrt{x^2-1}}\right)\:dx$? So what I've tried is substituting $x\:=\:\frac{1}{\sin t}$. So then I'll have that when $x\rightarrow ...
0
votes
1answer
33 views

Definite integral - Integration by parts [closed]

Let $p,f,g,q$ be continuous functions on $[a,b]$. How can I show that $$\int_a^b (pf'g'+qfg)dt=\int_a^b f(-(pg')'+qg)dt$$ Maybe by integrations by parts?
4
votes
2answers
170 views

How can I calculate $ \lim_{h\to 0} \frac{1}{h}\int_{2}^{2+h} F(x)\,dx$?

Let, say, $F(x) = \sin(x^2)$ which is continuous, therefore there exists a $c \in [2,2+h]$ such that $$ F(c) = \frac{1}{h}\int_{2}^{2+h} F(x)\,dx.$$ I'm trying to calculate the limit when $h$ goes ...
1
vote
3answers
134 views

Evaluate $\int_{-\infty}^{\infty} \frac{(1-\cos { y } )}{\mid{y}\mid^{1+\alpha}}dy$ [on hold]

How do I evaluate the following integral? $$\int_{-\infty}^{\infty} \frac{(1-\cos { y } )}{\mid{y}\mid^{1+\alpha}}dy=\frac{\pi}{\Gamma(1+\alpha)\sin(\frac{\pi\alpha}{2})}$$ Thank you in advance. ...
0
votes
0answers
62 views

What is $\int_{-1}^0\sin(t)e^{t^4}\mathop{\mathrm{d}t}$? [closed]

$\int_{-1}^0\sin(t)e^{t^4}\mathop{\mathrm{d}t}$ Is there a way of solving this definite integral using simple methods?