Questions about the evaluation of specific definite integrals.

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2
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1answer
47 views

Possibility of a closed form for $I_n = \int \frac{x^n e^{\tan^{-1}{x}}}{\sqrt{x^2+1}}\text{d}x$ where $n$ is a given integer.

The case of $n=2$ possesses an elementary closed form, which Mathematica 9 failed to find. This gives me an inkling of hope for the specific cases of the general form, as I have determined the ...
5
votes
1answer
61 views

Strange behavior of infinite products $\prod^{\infty}_{n=1} \ln (1+ \frac{1}{n} )^n$ and $\prod^{\infty}_{n=1} \ln (1+ \frac{1}{n} )^{n+1}$

There are two expressions marking the lower and upper bounds for number $e$: $$\left(1+\frac{1}{n} \right)^n \leq e \leq \left(1+\frac{1}{n} \right)^{n+1}$$ Naturally, I wanted to know if infinite ...
0
votes
2answers
55 views

How would you calculate $(200\int_0^\infty e^{-0.8t}-e^{-1.8t}\,dt)/(250\int_0^\infty e^{-0.8t} \,dt)$?

$$\frac{200\int_0^\infty e^{-0.8t}-e^{-1.8t} \, dt}{250\int_0^\infty e^{-0.8t} \, dt}$$ I am confused as to how you would integrate the e's from zero to infinity. What steps would you take? By the ...
2
votes
1answer
43 views

Closed form or simplification of a multiple definit integral of a product of a weight averaged parameters

I am trying to obtain a closed form solution of this definite integral, or in a form at least which simplify its numerical treatment. $$\int_{x_1=0}^1...\int_{x_N=0}^1 \prod_r \left( \frac {x_r f_r} ...
2
votes
3answers
68 views

Calculate $\int_0^1 \ \int_0^1 \ x \sin \lvert x^2-y^2 \lvert \; dx \; dy$

$$\int_0^1 \ \int_0^1 \ x \ \sin \lvert x^2-y^2 \lvert dx \ dy $$ $$\int_0^1 \frac{1}{2} \Big[ \sin \lvert x^2-y^2 \lvert \Big]_0^1 \ dy= \int_0^1 \frac{1}{2} \Big( \sin \lvert 1-y^2 \lvert - ...
2
votes
1answer
36 views

Finding Fourier series constant and integral

I have been studying Griffith's Intro to Electrodynamics. I am studying differential equations and Fourier series. I am studying the problem discussed here: Why is this allowed? ("Fourier's ...
1
vote
0answers
31 views

Is there a close form expression for the integral $ \int_a^b |x-c|^n e^{-x^2/2} $

Is there a close form expression for the integral \begin{align} \int_a^b |x-c|^n e^{-x^2/2} dx \end{align} by close form I mean it can be in terms of well know functions such as $Q$-function, ...
1
vote
3answers
75 views

Solving integral $\int _n^{2n}\frac 1x dx$ [on hold]

I want to solve the integral- $$\int _n^{2n}\frac 1x dx$$ From wolfram alpha I get it is $=log(2)\approx 0.69315$. But I am unable to solve it step by step.So I need help. Also,if the ...
-2
votes
2answers
90 views

Evaluate $\int_{-\infty}^{\infty}x^2 e^{-\alpha x^2+\beta x}dx$ [on hold]

As a consequence of this Q, I need some help evaluating the following integral: $$\int_{-\infty}^{\infty}x^2 e^{-\alpha x^2+\beta x}dx$$ Integration by parts wouldn't simplify things and I guess that ...
4
votes
1answer
39 views

How would you integrate this?

If we had the following integral: $$\int_{a}^{b} {\big(1+x^2 \big)^s} \space dx$$ Where $s$ is not given. Is there any general formula for this integration that works for all $s\in \mathbb{R}$?
1
vote
0answers
23 views

Difference Equations and Discrete Integral and Derivative

So I'm trying to learn difference equations, and the book that I'm using defines the following: The discrete derivative of a function $a_n$ of the integers is defined as: $$ D a_n = a_{n+1} - a_n $$ ...
1
vote
1answer
29 views

Is this polar equation correct? [on hold]

Find the area of the region bounded by: $$r=5\cos(10\theta),~~~~~ 0 \leq \theta \leq 2\pi$$ When I did this, I got $\frac{1}{2\sin(20\pi)}-\frac{1}{2\sin(0)}$ getting $0$, is this correct?
0
votes
2answers
41 views

$\int\limits_0^1 {\left( {1 - 2{x^2}} \right)f\left( x \right)dx}<0$, when $f$:convex and differentiable with $f(0)=0$

Let $f:[0,1]\to\mathbb{R}$ be a differentiable function that is convex and $f(0)=0$. Prove that: $\int\limits_0^1 {\left( {1 - 2{x^2}} \right)f\left( x \right)dx}<0$. I thought that since ...
1
vote
6answers
109 views

Integrate $\int_{-\infty}^\infty xe^{-\alpha x^2+\beta x}dx$ [duplicate]

I am familiar with the gauusian integral $$\int_{-\infty}^\infty e^{-\alpha x^2+\beta x}dx=\sqrt{\frac{\pi}{\alpha}}e^{\beta^2/(4\alpha)}$$ Could anyone help me to find out the value of the following? ...
0
votes
1answer
44 views

Limit with number of integrals tending to infinity

Let $F_0(x) = \ln x$. For $n \geq 0$ and $x >0$, let $F_{n+1}(x)=\int_0^x F_n(t)dt$. Evaluate $$\lim_{n \to \infty} \frac{n! F_n(1)}{\ln n}$$ Because the final intergal is from $0$ to $1$, I ...
0
votes
1answer
26 views

What would be a good cartesian equation to represent the shape of a wine glass?

I want to find the volume of a wine glass by using either the disk or shell method (solids of revolutions). The wine glass doesn't have to be of any particular dimensions, however it should roughly ...
0
votes
0answers
11 views

Closed-form of spherical expansion of Legendre polynomial $P_k(\sin{\theta}\cos{\varphi})$

During the times of working on some problem in astro/geophysics I have come across a problem involving an expansion into spherical harmonic functions (this is the remnance of nomenclature there used ...
1
vote
1answer
24 views

Convolution using Integration

Using integration, how would I solve f(t) convolve g(t) given that $$f(t)=u(t)-u(t-5)$$ and $$g(t)=2[u(t)-u(t-1)]$$ I know it should be $$\int_0^6 f(\tau) \ast g(t-\tau)~ d\tau = ...
1
vote
2answers
71 views

Integrating $\int^1_0 \dfrac{x^2e^{\arctan x}}{\sqrt{x^2+1}}$

This is a very hard integral that I am trying to solve. I’ve tried many substitutions, integration by parts, but I cannot evaluate this. Are there any other approaches I can take to solve this ...
1
vote
3answers
22 views

Laplace transform for $-t\cos(2t)$

This Laplace transform exercise is giving me a headache. I was trying to use the definition of the Laplace transform but when I make the $u$ and $dv$ substitutions for the integration by parts I never ...
4
votes
4answers
99 views

Evaluation of $\,\displaystyle \lim_{n\rightarrow \infty}\sum_{k=1}^n\sin \left(\frac{n}{n^2+k^2}\right)$

Evaluation of $\displaystyle \lim_{n\rightarrow \infty}\sin \left(\frac{n}{n^2+1}\right)+\sin \left(\frac{n}{n^2+2^2}\right)+\cdots+\sin \left(\frac{n}{n^2+n^2}\right)$ $\bf{My Try::}$ We can ...
2
votes
1answer
185 views

Calculate $I=\int_0^{1}\frac{1+x}{x^2+x+1}\log\left({\frac{x}{1-x}}\right)\,\mathrm dx$ without using complex analysis

Calculate $$I=\int_0^{1}\frac{1+x}{x^2+x+1}\log\left({\frac{x}{1-x}}\right)\,\mathrm dx$$ without using complex analysis. How to calculate without using the residue theorem? The correct answer ...
2
votes
3answers
89 views

Integral of $\int_{\mathbb{R}}e^{-\frac{x^{2}}{2}}\left(\cos\left(\pi nx\right)\right)dx$ [duplicate]

I was in need to urgently solve this integral. I already know the result in the closed form, does anybody know how to solve it? \begin{equation} ...
2
votes
3answers
82 views

How do I finish this trig integral $\int_0^{\pi/4}\frac{\sin^2 \theta}{\cos \theta}d\theta$?

I got up to the part where it's $$\frac{9}{125}\int_0^{\large \frac{\pi}{4}}\frac{\sin^2\theta}{\cos\theta}\,\,d\theta$$ but I can't figure out how to finish it off. By the way the original problem ...
2
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0answers
53 views

I am trying to show an inequality involving the product of three inner product terms

Define the inner product $\langle\cdot,\cdot\rangle$ for continuous functions defined on $[0,1]$ as: $$\langle\,f\mid g\rangle=\int_{0}^{1}f(x)g(x)e^{\rho x}dx,$$ where $\rho$ is a real number. I ...
2
votes
1answer
172 views

Another way of doing integration

What's your option for calculating this integral? No full solution is necessary, it's optional as usual. Calculate $$\int_0^1 \frac{2 \zeta (3)\log ^3(1-x) \text{Li}_2(1-x) }{x}-\frac{2 \zeta (3) ...
0
votes
1answer
91 views

Solution of this definite integral?

I want to find the expression for the following integral $$\int_0^\infty\text{d}x\frac{e^{i k x}}{x}$$ I have tried deriving with respect to $k$, transforming into an integral over the whole real ...
3
votes
3answers
423 views

Integration with a constant “a”: $ \int_0^a \frac1{\sqrt {a^2-x^2}} dx $

Find the exact value of $$ \int_{0}^{a} \frac{1}{\sqrt {a^2-x^2}} \mathrm {dx} $$ Where, $a$ is a positive constant Hi, guys can give me tips to solve this ? Should we use like u substitution?
1
vote
1answer
30 views

Verify that $Γ(x)$ = $(x − 1)Γ(x − 1)$ for all $x > 1$.

$Γ(x)$ = $\int_0^{∞} e^{-t}t^{x-1}dt$ Plugging $(x-1)$ into this equation, I get $Γ(x-1)$ = $\int_0^{∞} e^{-t}t^{x-2}dt$ Integrating by parts, I eventually end up with $-e^{-t}t^{x-1}]_0^∞$ + ...
0
votes
1answer
24 views

Find the volume of the solid generated by the region [closed]

Find the volume of the solid that is generated when the region enclosed by $ y = \cosh 2x, y = \sinh 2x, x = 0, $ and $ x = 5 $ is revolved around the x-axis.
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0answers
27 views

Piecewise function evaluation, using integration. [closed]

So i have this question and im completely lost. can someone help me please! I tried the first question, but its not correct, my answer was $g(-3) = 0$ and so it $g(3) = 0$ but apparently, i did ...
1
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2answers
79 views

How to integrate $\int\limits_0^\infty e^{-a x^2}\cos(b x) dx$ where $a>0$

How to integrate $$\int\limits_0^\infty e^{-a x^2}\cos(b x) dx$$ where $a>0$ The real problem is this integral $$\lim\limits_{\alpha\rightarrow 2}\int\limits_0^\infty e^{-a x^\alpha}\cos(b x) ...
0
votes
0answers
49 views

How to integrate a function like $(x(1-x))^{-1/3}$?

How to integrate a function like $(x(1-x))^{-1/3}$ within certain limit like $0$ to $1$ ? This question must be having duplicates here but I can't find.Feel free to close if there is a duplicate.
0
votes
0answers
28 views

Use convolution theorem to evaluate $\int_0^\infty e^{-((|a+su|)/c)^b}e^{-(u/k)^p}du$

$$\int_0^\infty e^{-((|a+su|)/c)^b}e^{-(u/k)^p}du$$ I cannot figure out what to do to solve a case like this, where the variable $u$ is only supported from $0$ to $\infty$. Some further information: ...
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0answers
13 views

Show that $\int_0^\infty \!e^-{(|a+sx|^bc^p+x^ps^b)/(c^ps^b)} \, \mathrm{d}x=\int_0^\infty \!e^-{(|a+sx|^b+c^p+x^p+s^b)/(c^p+s^b)} \, \mathrm{d}x$

I have been told this, and it seems to fit with what I would expect given certain values of b and p, but I just can't get from one to the other. Thanks.
-1
votes
1answer
43 views

Evaluating the following definite integral calculus

Given the following definite integral $$\int_0^4 \left[\left(1/2x^2 - 2x +8\right)-\left(1/4x^2+x\right)\right]\;\mathrm dx$$ I have done in the following process. $$\int_0^4 \left[\left(1/2x^2 - 2x ...
1
vote
2answers
31 views

Find volume of these solids using integration

a) The $(x>0, y< -1)$ region of the curve $y= -\frac{1}{x}$ rotated about the $y$-axis. The instructions say that one should use the formula: $V = \int 2πxf(x) dx$ I used another method and ...
0
votes
1answer
53 views

Having Troubles With This Integration Problem

The question I'm having troubles with is as follows: Evaluate $\int_{-r}^r\sqrt{r^2-t^2}\,dt$ (Hint: substitute $t=r\sin x$) So, immediately I did $dt=r\cos x\,dx$ and substitute it all in... ...
0
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0answers
33 views

Leibniz integral rule definition

https://en.wikipedia.org/wiki/Leibniz_integral_rule If we have an integral $$\int_{y_0}^{y_1} f(x, y) \,\mathrm{d}y$$ then for $x$ in $(x_0, x_1)$ the derivative of this integral is thus ...
0
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0answers
15 views

Relationship between Lippmann-Schwinger integrals of different dimensions

Define $G_n (\mathbf{x},\mathbf{x}')$ as $$ G_n (\mathbf{x},\mathbf{x}') = \lim_{\epsilon \to 0^{+}} \left[\dfrac{1}{(2 \pi \hbar)^{n}} \int_{\mathbb{R}^{n}} \mathrm{d}^{n}\mathbf{p} \dfrac{e^{i ...
0
votes
0answers
78 views

I am a Math Hobbyist. I have made some simple discoveries in Math. How do I share it with the Math community out there? [closed]

I am a Computer Engineering graduate and have taken many courses in Math of course. While I was in the University, I got myself lost in the world of mathematics and I discovered stuff that I felt ...
0
votes
1answer
23 views

Finding volume of a solid of revolution

I need to find the volume of the solid that is formed when the (x>0, y< -1) region of the curve y= -1/x is rotated about the y-axis. If I'm correct, this volume can be calculated by: Evaluating ...
5
votes
2answers
149 views

Improper Integral $\int_0^1\frac{\arcsin^2(x^2)}{\sqrt{1-x^2}}dx$

$$I=\int_0^1\frac{\arcsin^2(x^2)}{\sqrt{1-x^2}}dx\stackrel?=\frac{5}{24}\pi^3-\frac{\pi}2\log^2 2-2\pi\chi_2\left(\frac1{\sqrt 2}\right)$$ This result seems to me digitally correct? Can we prove ...
2
votes
1answer
53 views

Closed form of an integral $\int_0^{\pi/2} \ln^n (\sin x) \, dx$

Let $n \in \mathbb{N}$. May we have a closed form for the integral: $$\mathcal{J}=\int_0^{\pi/2} \ln^n (\sin x) \, {\rm d}x$$ One obvious approach would be to go through beta functions and ...
3
votes
1answer
54 views

Compute $\int_0^1 \frac{ 1}{1 + x^{1/2}}\,dx$. [closed]

Basically, the question is $$\int_0^1 \frac{1}{1+x^{1/2}}\,dx.$$ I have no idea how to approach this and have spent hours to no avail. Any help would be gladly appreciated. Thanks!!
1
vote
1answer
40 views

Definite integral of a continued fraction function

I came up with this function written as the following continued fraction (please correct me if my notation is incorrect): for $n\in\mathbb{N}$, let $$f(x;n)=x+\operatorname*{\LARGE ...
7
votes
5answers
469 views

Solve the following trigonometric integral [closed]

Calculate: $$\int _{0}^{\pi }\cos(x)\log(\sin^2 (x)+1)dx$$
2
votes
1answer
73 views

The closed form of $\int^\infty_{B}e^{-(x+\frac{A}{x})}\,dx$, where $A>0$, $B>0$.

What tools, ways would you propose for getting the closed form of this integral? $$\int^\infty_{B}e^{-\left(x+\frac{A}{x}\right)}\,dx,$$ where $A>0$, $B>0$. When $B=0$, from Table of ...
2
votes
5answers
110 views

Quick integral question

Sorry about the formatting, but how would I go about this question: $$\frac{d}{dx} \int_{\cos x}^1 \sqrt{(1 + e)^t} dt$$ What I've learned in class is that the derivative of an integral is just the ...
5
votes
2answers
126 views

Difficult Integral $\int_0^{1/\sqrt{2}}\frac{\arcsin({x^2})}{\sqrt{1+x^2}(1+2x^2)}dx=$

I have a difficult integral to compute.I know the result, but need to know the method of calculation. How prove this result? ...