Questions about the evaluation of specific definite integrals.

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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 ...
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0answers
10 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 ...
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0answers
9 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 = ...
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1answer
48 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 ...
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3answers
21 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 ...
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4answers
87 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
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1answer
155 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 ...
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3answers
78 views

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

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
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3answers
75 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 ...
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0answers
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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 ...
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0answers
113 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) ...
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1answer
86 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 ...
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3answers
418 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?
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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 $-e^{-t}t^{x-1}]_0^∞$ + ...
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1answer
23 views

Find the volume of the solid generated by the region [on hold]

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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27 views

Piecewise function evaluation, using integration. [on hold]

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 ...
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2answers
71 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) ...
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0answers
47 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.
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0answers
27 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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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.
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1answer
42 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 ...
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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 ...
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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... ...
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0answers
32 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 ...
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0answers
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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 ...
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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? [on hold]

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 ...
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1answer
22 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
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2answers
143 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
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1answer
52 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
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1answer
54 views

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

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
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1answer
37 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 ...
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5answers
458 views

Solve the following trigonometric integral [on hold]

Calculate: $$\int _{0}^{\pi }\cos(x)\log(\sin^2 (x)+1)dx$$
2
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1answer
71 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 ...
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5answers
108 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
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2answers
123 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? ...
2
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2answers
66 views

Prove that $\int_{0}^\pi x^{2k} \cos(h x) dx\geq 0$.

Prove that $$\int_{0}^\pi x^{2k} \cos(h x) dx\geq 0$$ for all $k\in\mathbb N$ and $h$ even integer. I have tried with Induction Principle (for $h$) but without success.
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4answers
62 views

Give that $f$ is a decreasing continuous function and that $f(x+y) = f(x) + f(y) -f(x)f(y)$ and $f'(0)=-1;$ Then find $\int_{0}^{1}f(x)dx$

Give that $f$ is a decreasing continuous function and that $$f(x+y) = f(x) + f(y) -f(x)f(y)$$ and $f'(0)=-1;$ Then it is to be found what is $\int_{0}^{1}f(x)dx$ I am at a loss on how to approach ...
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2answers
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meaning of definite integral

So to my knowledge a definite integral's significance is how it shows the "intensity" or area under the curve for a function. However, I am confused then why the definite integral for x from 0 to 1 ...
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Riemann-Stieltjes Integral Substitution

I want to prove $\int^b_a\,f(g(x))\,dg(x) = \int^{g(b)}_{g(a)}\,f(x)\,dx$ for all f continuous. Firstly, $\int^b_a\,f(g(x))\,dg(x) = \int^b_a\,f(g(x))g'(x)\,dx$, since g is continuous and ...
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0answers
85 views

tough definite integral: $\int_0^\frac{\pi}{2}x\ln^2(\sin x)~dx$

Any ideas on $\int_0^\frac{\pi}{2}x\ln^2(\sin x)\ dx$ ? Best numerical approximation I can get is $0.2796245358$ Is there even a closed form solution?
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1answer
93 views

Closed form for this integral $I=\int_0^{1}\frac{{\arcsin}({x^2})}{\sqrt{1-x^2}}dx$

I’m trying to find a closed form for this integral.Any help is appreciated.Thanks $$I=\int_0^{1}\frac{{ \arcsin}({x^2})}{\sqrt{1-x^2}}dx$$
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0answers
36 views

Meijer function as a product of two Meijer's functions

I want to evaluate an integral $I_1$ defined in $Eq.(1)$ as \begin{align} I_1=\int_{0}^{\infty}\frac{x\exp(-\beta x)K_1(\alpha x)}{1+x}dx\tag{1} \end{align} Where $K_1(.)$ is modified first order ...
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1answer
47 views

Convert Riemann sum to definite integral: $\sum_{i = 1}^n \frac{n}{n^2 + i^2}$

I am having trouble with this problem. Basically, I am given a Riemann sum and I have to rearrange it so that I can deduce the definite integral that it is equivalent to. Thank you. $$\lim_{n \to ...
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4answers
63 views

Using trig identities to evaluate $\int_{0}^{\pi/2} \sqrt{1-\sin x} \, dx$

Use the identities $$\cos 2x=2\cos^2 x -1=1-2\sin^2 x$$ $$\sin x=\cos \left(\frac{\pi}{2}-x\right)$$ to help evaluate $$\int_{0}^{\pi/2} \sqrt{1-\sin x} \; dx$$ I've already done ...
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0answers
17 views

Can an iterated integral over a box R ={(x,y,z)|x∈[0,a], y∈[0,b], z∈[0,c]} be expressed in eight different ways?

this is my first time on stack exchange so sorry if I am not following any guidelines. I received this exact question on a midterm and answered yes, it is possible, which was considered wrong on the ...
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0answers
27 views

Integral in vector form

I have two vectors $x_{1}$ and $x_{2}$ and some function $f(x, k)$. So for example function $f$ can be evaluated at some point say $f(x_{1}, 10)$. Then I have an integral written ...
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4answers
99 views

I want to solve $\int \frac{2}{x^2(x^2+1)^2}dx$

I want to solve this primitive $$I=\int \frac{2}{x^2(x^2+1)^2}dx.$$ I substitute $u=x^2$ then, $$I=\int \frac{2}{x^2(x^2+1)^2}dx=\int \frac{du}{u^{3/2}(u+1)^2}=\cdots$$ How do I use partial fraction ...
0
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1answer
52 views

How do I evaluate $\displaystyle \int_{-\infty}^z e^{\frac{-t^2+2t\alpha\mu}{2\sigma^2\alpha^2}+\frac{\lambda t}{1-\lambda}} dt$ ??

How do you evaluate: $$\displaystyle \int_{-\infty}^z e^{\frac{-t^2+2t\alpha\mu}{2\sigma^2\alpha^2}+\frac{\lambda t}{1-\alpha}} dt = ??$$ Many thanks.
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5answers
78 views

How can I solve $\int \frac{3x+2}{x^2+x+1}dx$

I want to compute this primitive $$I=\int \frac{3x+2}{x^2+x+1}dx.$$ I split this integral into two part: $$\int \frac{3x+2}{x^2+x+1}dx=\int \frac{2x+1}{x^2+x+1}dx+\int \frac{x+1}{x^2+x+1}dx,$$ For ...
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0answers
48 views

Help evaluating the integral of $\frac{1}{1+\sqrt{1-x^2}}$ over $[0,1]$ [closed]

Can someone help me with this please: $$ \int_0^1 \frac{dx}{1+\sqrt{1-x^2}} $$ I substituted putting $u= \sqrt{1-x^2}$ and I got to this $$ \int_0^1 \frac{u\,du}{(u+1)\sqrt{1-u^2}} $$ I don't know ...