Questions about the evaluation of specific definite integrals.

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Calculate integral

I have to calculate $$\operatorname{PV}\int_{-\infty}^{\infty}\frac{1}{\pi}\frac{y}{1+y^2}dy$$ I ended up with ...
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Feynman Integration Problem

$$ I = \frac{\pi^2}{8} - \int_0^1 \frac{\arctan(x)}{\sqrt{1-x^2}} \,dx $$ Evaluate $I$ $$ I = \frac{\pi^2}{8} - \int_0^1 \frac{\arctan(x)}{\sqrt{1-x^2}} \,dx$$ $$f(a) = \int_0^1 ...
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2answers
33 views

Definite integral calculation

$$\int_0^{2\pi}a^2(1-\cos(x))^2\sqrt{((a-a\cos(x))^2+\sin^2(x))}\,dx$$ After some work I get $$2a^3\int_0^{2\pi}(1-2\cos(x)+\cos^2(x))\sin\left(\frac x2\right)\,dx$$ And stopped:( Any ideas?
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0answers
37 views

Integral formulas involving continued fractions

Ramanujan posed the following formulas as questions in the Journal of Indian Mathematical Society: $$\int_{0}^{\infty}\dfrac{\sin nx\,\,dx}{{\displaystyle x + \dfrac{1}{x +}\dfrac{2}{x +}\dfrac{3}{x ...
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1answer
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Sophomore's dream changing “x”

"Sophomore's Dream" says $\sum_{n=1}^{\infty}n^{-n}=\int_0^1x^{-x}$ Can you replace the $x$ and $n$ with $2x$ or $x^3$ (and $2n$ or $n^3$) or something? I would guess not, because replacing $x$ with ...
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1answer
78 views

Prove $ \lim\limits_{n\to\infty}\int_0^1 f(x)g(nx)\,dx=\int_0^1 f(x)\,dx\int_0^1 g(x) \, dx $ [duplicate]

Let $f$ and $g$ be a real valued continuous functions on $\mathbb{R}$ such that $f(x+1)=f(x)$ and $g(x+1)=g(x)$ for all $x\in \mathbb{R}$. Prove that $$ \lim_{n\to\infty}\int_0^1 ...
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1answer
49 views

Questions about integration

I'm still a bit confused about definite integration although got the basic idea of how to do integration. The problem is to integrate functions on a uniform distribution over [50, 150]. Firstly ...
3
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2answers
90 views

use parseval's identity to evaluate the integral $ \int_{-\pi}^{\pi}\sin^4 xdx$

use parseval's identity to evaluate the integral \begin{equation} \int_{-\pi}^{\pi}(\sin x)^4dx\end{equation} I'm familiar with Parseval's identity which states that for each piecewise continuous ...
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0answers
40 views

Changing order of integration for the triple integral $ \int\limits_{0}^{2} \int\limits_{0}^{2z} \int\limits_{y}^{2y} f_{(x,y,z)}\; dx\, dy\, dz $

I need to change order of integration for the following triple integral: $$ \int\limits_{0}^{2} \int\limits_{0}^{2z} \int\limits_{y}^{2y} f_{(x,y,z)}\; dx\, dy\, dz $$ The domain of integration is ...
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Calculate $\int_{0}^{1} \frac{x^3}{\sqrt{1+x^4}}dx$ [on hold]

Help with this integral, please! $$\int_{0}^{1} \frac{x^3}{\sqrt{1+x^4}}dx$$
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4answers
64 views

Estimate from below $\int_0^\pi e^{-t}\cos nt dt$ without calculate it.

Estimate from below the following integral $$\int_0^\pi e^{-t}\cos nt dt$$ without calculate it. Here $n\in\mathbb N$. Any suggestions please?
4
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0answers
33 views

Solving integral with spherical bessel functions

I would like to find if possible a solution (closed form) for the following integral: $$\frac{1}{2 \pi}\cdot\int\limits_0^{2\pi}\exp\bigg[-ia(\cos x+\sin x)\bigg]\,j_{0}(b\cos x)\,j_{0}(b\sin ...
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0answers
34 views

A Combinational identity using permutations

For a distribution {$p_1,p_2, …,p_m$}, with $p_i>0$ and$\sum_1^m{p_i}=1$ , let $J$ be a subset of size $j$, and $m>j\geq1$. It holds that: $$\int_0^1\prod_{i \in J} (x^{-p_i}-1) dx = ...
6
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2answers
35 views

Definite integral of product of functions

I know it's not correct to write: $$\int_{a}^{b}f(x)g(x) dx = \int_{a}^{b}f(x)dx\int_{a}^{b}g(x)dx$$ This result seems obvious, but I can't think of a way to prove that $\int_{a}^{b}f(x)g(x) dx$ ...
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1answer
20 views

What happens when the interval of an integral changes from infinity to a constant number?

There exist a calculation about electromagnetic mass: $$m_\mathrm{em} = \int {1\over 2}E^2 \, dV = \int\limits_{r_e}^\infty \frac{1}{2} \left( {q\over 4\pi r^2} \right)^2 4\pi r^2 \, dr = {q^2 \over ...
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0answers
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Closed form of an infinite series of integrals $\int_{0}^{\eta} \cos nt \cos t \sqrt{\cos^2 t - \cos^2 \eta}$

Let $$ I(n,\eta) = \int_{0}^{\eta} \cos nt \, \cos t \, \sqrt{\cos^2 t - \cos^2 \eta}\; dt $$ where it is known that $0 < \eta \leq \frac \pi 2$. Is it possible to evaluate $S$, the infinite ...
13
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3answers
191 views

Integral involving Clausen function ${\large\int}_0^{2\pi}\operatorname{Cl}_2(x)^2\,x^p\,dx$

Consider the Clausen function $\operatorname{Cl}_2(x)$ that can be defined for $0<x<2\pi$ in several equivalent ways: ...
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1answer
45 views

Changing argument into complex in the integral of Bessel multiplied by cosine

I got a problem solving the equation below: $$ \int_0^a J_0\left(b\sqrt{a^2-x^2}\right)\cosh(cx) dx$$ where $J_0$ is the zeroth order of Bessel function of the first kind. I found the integral ...
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0answers
48 views

Double Integration Working Help

Help I dont know how to approach this question, I have the answer but dont know how to write a detailed working process of obtaining it. It is supposed to find the surface area of a cone that is $z = ...
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2answers
63 views

what will be the value of this integral

$$ \large{ \int^{\Large{\frac{\pi}{2}}}_{0} \left[ e^{\ln\left(\cos x \cdot \frac{d(\cos x)}{dx}\right)} \right]dx}$$ We know that $\large{a^{log_a(c)} = c}$. But in this question, the expression in ...
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1answer
17 views

Establishing a Variant of the Mean Value Property of Harmonic Functions

Let $u:U\to \mathbb{C}$ be harmonic and $\overline{D}(P,r)\subset U$. Verify the following variant of the mean value property of harmonic functions: $$u(P)=\frac{1}{2\pi r}\int_{\partial ...
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The Laplace transform of $\frac{\ln(1+at)}{1+t}$

By expressing the square of the exponential integral as a double integral and then making a change of variables, one can show $$ \int_{0}^{\infty} e^{-2zt} \ \frac{\ln(1+2t)}{1+t} \, dt = \frac{e^{2z} ...
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1answer
15 views

Limit of Cosine and Sine Fourier Transforms

If I define the cosine and sine Fourier transform as (skipping constant prefactors $(2\pi)^{0.5}$): $$\mathcal{F}_C\{f(x)\}=\int_0^{\infty}\,f(x)\,\cos(\omega x)\,dx$$ and ...
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+50

Functional Analysis, a question that needs clarification.

Find the norm of the linear operator $A:C[-1,1]\to L^p[-1,1]; p\geq1$ that is defined as: $$A(x(t))=\int_{-1}^{1}{{x(s)\over (s-t)^{1 \over 3}}}ds$$ Can someone provide an answer with a little more ...
0
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0answers
38 views

MAple 17 won't evaluate my integral

I type this into maple and it won't evaluate it: $$ \int_{0}^{1}\pi ((-y^4+1)^2-(1-y)^2) dy $$ I've also tried $$ evalf(\int_{0}^{1}\pi ((-y^4+1)^2-(1-y)^2) dy)$$ It just returns for both cases $$ ...
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2answers
43 views

Definite integral from -1 to 0 [on hold]

How would I evaluate this definite integral $$ \int_{-1}^{0}\tan x dx- \int_{-1}^{0}\sin^2 x dx $$ All i need to know is what to do when an integral is on an interval of -1 to 0. I could do this ...
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0answers
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Strange triple integral of an inverse function

Let $$ \Omega(a, b, c) = \min\left\{\theta\ge0\ \text{s.t.}\ \tan(a\theta) + \tan(b\theta) + \tan(c\theta) = 1\right\} $$ What is the value of the following integral $$ I = ...
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1answer
35 views

Definite integral including natural log, cosine, and hyperbolic sine

Here is an integral question I have, I am solving some other problems like this but I am stumped on this one: $$\int_0^{\pi+1}\frac {\ln(\cos(x+1))}{\sinh(x^2)}dx$$ I used some methods such as ...
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2answers
161 views

Definite integral, $\frac 1{\ln(x)}$

What is $$\int_0^{\pi^{2}}\frac 1{\ln(x)}dx$$ I tried using complex residues and some identities, but no luck. Any suggestions?
3
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2answers
79 views

Computing $\int_{0}^{+\infty}\frac{\log(x)}{\sqrt x(1+{x^2})}dx$.

I would like to compute the following integral : $$\int_{0}^{+\infty}\frac{\log(x)}{\sqrt x(1+{x^2})}dx$$ using Residue theorem. I took the contour corresponding to half of the "donuts" ...
3
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2answers
38 views

Evaluating a limit involving a definite integral

I want to prove the following limit evaluates to $0$ without using any techniques that involve complex numbers. I already solved it using residues and it's pretty straight forward, but it feels rather ...
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0answers
68 views

How to find the value of this integral?

This integral to the value \begin{align} \int_0^1\frac{\ln^2(1+x)\ln^2 x}{1-x}\ dx=&\ ...
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3answers
99 views

Trigonometric Substitution in $\int _0^{\pi/2}{\frac{ x\cos x}{ 1+\sin^2 x} dx }$

Evaluate $$ \int _{ 0 }^{ \pi /2 }{ \frac { x\cos { (x) } }{ 1+\sin ^{ 2 }{ x } } \ \mathrm{d}x } $$ $$$$ The solution was suggested like this:$$$$ SOLUTION: First of all its, quite obvious to have ...
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0answers
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$\int_{-\pi/2}^{\pi/2} \cos(a \cos\theta) e^{im\theta} e^{-ib\sin\theta} \mathrm{d}\theta $ Integration

I am struggling to find the integration of the expression below, $$\int_{-\pi/2}^{\pi/2} \cos(a \cos\theta) e^{im\theta} e^{-ib\sin\theta} \mathrm{d}\theta $$ where $a$ and $b$ are arbitrary constant ...
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1answer
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Asymptotic of a complex integral

Consider the following integral $$f(x):=\int_x^{+\infty}re^{-(r+ir^2)}dr$$ I want to understand the asymptotic behavior of $f(x)$ as $x\rightarrow +\infty$ Thank you for any suggestion.
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1answer
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$\int_{0}^{\frac{\pi}{4}} e^{\sec x} \frac{\sin( x + \frac{\pi}{4})}{(1 - \sin x) \cos x}\, dx$?

How do I find the value of $$ \int_{0}^{\frac{\pi}{4}} e^{\sec x} \dfrac{\sin\Big( x + \dfrac{\pi}{4}\Big)}{(1 - \sin x) \cos x} \;\mathrm{d}x $$
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2answers
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I was Stumped by this Challenging Integration/Limit Problem

Can someone show me how I do the following: \begin{align} I_n & = \int_0^1 \sqrt{ \frac 1 x + n^2 x^{2n} } \ \mathrm dx \\[8pt] & \lim_{n\rightarrow \infty } I_n = \text{ ?} \end{align}
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1answer
55 views

Help with an Inverse Trigonometry Integral 2

Evaluate $$\int^{1/{\sqrt{3}}}_{-1/{\sqrt{3}}} \frac{x^4}{1-x^4}\cos^{-1}\frac{2x}{1+x^2} \mathrm{d}x\\= \frac{\pi}{a}\ln(b+\sqrt{c}) +\frac{\pi^{d}}{e} - \frac{\pi}{\sqrt{f}}$$ Then Find ...
2
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1answer
76 views

Integral with Logarithms

$$\displaystyle \int _{ 0 }^{ \pi /2 }{ \log(\cos(x))\log(\sin(x)) \ dx } = \dfrac { \pi { \ln}^{ A }(B) }{ C } -\dfrac { { \pi }^{ D } }{ E } $$ $$$$ This was one solution, but it went completely ...
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1answer
42 views

Computing $\int_{0}^1 \frac{\left( (1-x )a +x b\right)^2}{(1-x)c +x d} dx$

I want to find the integral of \begin{align*} \int_{0}^1 \frac{\left( (1-x )a +x b\right)^2}{(1-x)c +x d}dx \end{align*} for any $a,b$ and $c>0$ and $d>0$. Using Wolfram-Alpha I found that ...
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2answers
115 views

Cannot understand an Integral

$$\displaystyle \int _{ \pi /6 }^{ \pi /3 }{ \frac { dx }{ \sec x+\csc x } } =\frac { \sqrt { a } -b }{ 2 } +\frac { \sqrt { c } }{ 2 } \log(\sqrt { d } +\sqrt { e } -\sqrt { f } -g)$$ I had to solve ...
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Limit integral w/ sine function

Here's a random problem I thought of: $$\lim_{\alpha\to 0}\int_0^{\infty}\sin(\alpha x) \;\mathrm{d}x$$ What I'm trying to create is a sine function with an "infinite period", meaning instead of the ...
5
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1answer
43 views

Fourier transform in three dimensions getting out of hand

I have the following integral I wish to compute, it transforms a quantum position wave function into momentum space: $$\phi(\mathbf p)=\int\frac{\mathrm d^3r}{(2\pi\hbar)^{3/2}}e^{-i\mathbf{p\cdot ...
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0answers
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Double Integral to evaluate volume over region [closed]

I'm not sure how to write the integral needed for this problem: Find the volume of the solid bounded by the graphs of the equations: $$x^2 + z^2 = 1 $$ $$y^2 + z^2 =1 $$ And the first octant. I ...
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1answer
40 views

Calculate: $I=\int_2^5 \frac{\ln(x^2+1)}{x}dx$ [closed]

Calculate: $I=\int_2^5 \frac{\ln(x^2+1)}{x}dx$ I used wolframalpha.com and get the result: $I=\dfrac{Li_2(-4)-Li_2(-25)}{2}$ Who can find this result's representation with primary function?
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3answers
138 views

A curious equation containing an integral $\int_0^{\pi/4}\arctan\left(\tan^x\theta\right)d\theta=\frac{\ln2\cdot\ln x}{16}$

I came across an interesting problem that I do not know how to solve: Find $x>0$ such that $$\int_0^{\pi/4}\arctan\left(\tan^x\theta\right)d\theta=\frac{\ln2\cdot\ln x}{16}.$$ Could you ...
3
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0answers
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+50

Can these integrals be represented in closed form?

This paper in the formula F.3.6 (page 271) gives the following formula for the derivative of Hurwitz Zeta function: $$\frac ...
0
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1answer
64 views

Prove the integral $\int_0^\infty \frac{x^n e^{-\mu x}}{x+\beta} dx$

I am trying to prove this integral here: Where Ei is the exponential integral. Unfortunately I don't have the right result yet, but I have other result that is not for this case, but I think it can ...
0
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0answers
33 views

Need proof of integration of sine parametrized functions [duplicate]

Yesterday, i encountered an integral formula (actually it's a generalization, i think). This : $$\int_0^\pi x f(\sin x)\,dx = \frac{\pi}{2} \int_0^\pi f(\sin x)\,dx$$ For simple functions like ...
0
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1answer
17 views

Infinite Integral of a Bessel Function

I need to calculate the following integral $$ \int_0^{\infty}xdxJ_n(kx) $$ Integrating it by parts and using the normalization of Bessel functions, I find it (somewhat heuristically) to equal the ...