Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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12 views

Is there any trick for evaluate this integral?

Does the following function can be simplified or solved? $$R(i) = \int_{y\in S} {\frac{{w(y) g(y,i)_{}^\sigma }}{{\int_{x\in S} {h(x)g(x,y)_{}^\sigma f(x,y)_{}^\sigma dx} }}dy} $$ where S is a ...
1
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1answer
21 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 $-e^{-t}t^{x-1}]_0^∞$ + ...
3
votes
1answer
24 views

Solve the differential equation : $0.5 \frac{dy}{dx}=4.9-0.1y^2$

The question is to solve the differential equation : $$0.5 \frac{dy}{dx}=4.9-0.1y^2$$ What I have attempted: $$0.5 \frac{dy}{dx}=4.9-0.1y^2$$ $$ \frac{dy}{dx} = \frac{4.9-0.1y^2}{0.5} ...
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1answer
19 views

Find the volume of the solid generated by the region

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
22 views

Prove that the following function is convex?

I am trying to prove that the function $$g(\alpha)=\ln\Big(\sum_{i=1}^{n}t_i^\alpha+A(\alpha)\Big) ~~t_i, \alpha>0,$$ where $A(\alpha)=\int_{a}^{b}x^{\alpha+\beta}e^{-\lambda x^\beta}\,dx$,is ...
2
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1answer
55 views

Find $\int\limits^{\infty}_{0}\int\limits^{\infty}_{0}{\frac{1}{(x+y)^{3/2}}\exp\left\{-\frac{a^2}{2(x+y)}\right\}}\,dy\,dx$.

In my posterior probability computation, I got the following integration and I could not figure it out. ...
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0answers
21 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
21 views

Is it true in general that $\int_{|X| \leq \epsilon} |X|^r \, d\mathbb{P} \leq \epsilon^r$?

If I have that $X$ is a random variable, for $\epsilon > 0$, and $r \geq 1$, is it true that: $$\int_{|X| \leq \epsilon} |X|^r \, d\mathbb{P} \leq \epsilon^r.$$? If so, is there a reason why? ...
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0answers
30 views

Important numerator and denominators in the evaluation of the integral: $\int_0^\infty x^t \operatorname{csch} x\text{ d}x$

$$\int_0^\infty x^t\operatorname{csch}x\text{ d}x=\frac{a\zeta(t+1)}{b}$$ for $t\in\Bbb{N}$ How might one represent $a,b$ in terms of $t$? (Note that $a,b\in \Bbb{N}$) If possible, could one also ...
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2answers
34 views

Differential Equations $ v \frac{dv}{dx} = -g \frac{a^2}{x^2}$

Question: A particle is projected vertically upwards from the Earth's surface. Its distance $x$ from the centre of the Earth is connected with its upwards speed $v$ by the differential ...
-1
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3answers
61 views

Without computing, is the integral of $\int_0^1 t(t-1)(t-2)\,dt$ positive or negative? [on hold]

I have to graph the function, but I don't think I'm doing it right. Here is a picture of it Sorry, this is my first time using this site and I don't know how to use MathJax yet.
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0answers
11 views

Need help with $\int\mathrm{exp}[-C(\frac{1}{(x -\frac{1}{2})^2 + (x + \frac{1}{2})^2} -1)^S ] \mathrm{dx}$ for a statistical mechanics problem

Can someone help solving this integral?: $$\int\limits_{-\frac{1}{2}}^{0} \mathrm{exp}\left[-C \left( \frac{1}{(x -\frac{1}{2})^2 + (x + \frac{1}{2})^2} -1 \right)^S \right] \mathrm{dx}$$ The ...
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3answers
31 views

Integrate the following equation. (exponential function)

Integrate $$\frac{e^x -2}{e^{x/2}}$$ This is my calculation: but it is wrong....
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0answers
16 views

Riesz-Type Representation Theorems for Convex Functionals

It is well known that any positive linear functional $L$ on the spase $C_c([a,b])$ of functions continuous on an interval $[a,b]$ with compact support can be written as \begin{align*} ...
1
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0answers
29 views

Problem with $\int_{0}^{\infty} \frac{\log^2(x)}{1+x^2}$ (by residues) [duplicate]

I, I am trying solve the following integral $$\int_{0}^{\infty} \frac{\log^2(x)}{1+x^2}$$ Teachers teached me that I can solve the integral $$\int_{0}^{\infty} ...
0
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3answers
37 views

Area of a rectangle within a curve

The cargo space of a bulk carrier is 60m long. The shaded part of the diagram represents the uniform cross-section of this space. It is shaped like a parabola with equation ${{1\over 4}x^2, - 6 \le ...
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0answers
13 views

Combining two results from partial integration

I have a set of two PDEs: $$\partial_{\tau}\theta+\partial_{\eta}\psi=0$$ $$\partial_{\tau}\psi=-\partial_{\eta}\theta+\alpha\partial_{\eta}^{2}\psi$$ These can be combined into a wave equation of ...
1
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1answer
17 views

Computing a line integral where the curve is in polar coordinates

Compute $\int \limits_{C} F.dr$ for $F(x,y)=(y,x)$ and $C$ is the curve given by $r=1+\theta$ for $\theta \in [0,2\pi]$ My Attempt Am I correct in saying that $F$ is a conservative vector field ...
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1answer
32 views

solving $\int_0^{\pi/2} sin^2\theta \sqrt{1-k^2sin^2 \theta}d\theta $

I have the following integration to solve. $$f(k) = \int_0^{\pi/2} sin^2\theta \sqrt{1-k^2sin^2 \theta}d\theta $$ assuming $sin\theta = t$ which results $d\theta = \frac{dt}{\sqrt{1-t^2}}$ and when ...
1
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2answers
48 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) ...
1
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2answers
69 views

Find $\lim_{n \rightarrow \infty}\frac{1}{n} \int_{1}^{\infty} \frac{dx}{x^2 \log{(1+ \frac{x}{n})}}$

Find: $$\lim_{n \rightarrow \infty} \frac{1}{n} \int_{1}^{\infty} \frac{dx}{x^2 \log{(1+ \frac{x}{n})}}$$ The sequence $\frac{1}{nx^2 \log{(1+ \frac{x}{n})}}=\frac{1}{x^3 \frac{\log{(1+ ...
3
votes
3answers
52 views

Find $\lim_{n \rightarrow \infty} \int_0^n (1+ \frac{x}{n})^{n+1} \exp(-2x) \, dx$

Find: $$\lim_{n \rightarrow \infty} \int_0^n \left(1+ \frac{x}{n}\right)^{n+1} \exp(-2x) \, dx$$ The sequence $\left(1+ \frac{x}{n}\right)^{n+1} \exp{(-2x)}$ converges pointwise to $\exp{(-x)}$. So ...
1
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0answers
8 views

When is the Stieltjes integral of bounded variations?

I was trying to figure out when a Riemann or Lebsgue Stieltjes integral is of bounded variation. For simplicity let $f$ be a increasing RCLL function; when is that $$\int_0^t g(x) df(x)$$ is of ...
1
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1answer
45 views

Weird indefinite integral homework questions

I'm solving a couple of integration problems using the method of changing variables, and would like assistance with two particular problems that I can't seem to solve. I completed rest of the problems ...
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2answers
32 views

Evaluate the inverse trigonometric integral

Evaluate the integral:$\int_{1}^{2} \frac{\tan^{-1} x}{\tan^{-1} \frac {1}{x^2-3x+3}} dx$ On applying the property $\int_{a}^{b}f(x)dx=\int_{a}^{b}f(a+b-x)dx$ I dont seem to reach any where
1
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1answer
27 views

Evaluating the integral of a sine function

I am having some trouble with part (b) and part (c) of this: (b) I know that I have to differentiate it and I get $\cos (\frac{\pi}{x})$ and by using the definite integral I get $\cos (\pi n)-\cos ...
2
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1answer
53 views

If $y'+y=|x|$ and $y(-1)=0$, what is $y(1)$?

If $y'+y=|x|$ and $y(-1)=0$, what is $y(1)$? I calculated the integrating factor to be $e^x$. Then $e^x y'+ e^x y=e^x |x|$ hence $\frac {d(e^x y)}{dx}=e^x |x|$ hence $d(e^x y)=e^x|x|dx $ ...
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0answers
22 views

Definition of integrability for sequences

My text book does not provide much about counting measures and integration. So I decided to setup integration on space $(N , P(N) , \mu_c ,R)$ myself imitating the construction of Lebesgue integral. ...
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1answer
25 views

Finding length of curve $y^2 = 64(x+3)^3$ for $0 \le x \le 3$

Not getting the right answer for this, can someone point me to where I'm going wrong?
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0answers
31 views

How to prove the following questions by IBP? (Integrated By Parts) [on hold]

So this is the question that I have to solve. I know this is related to IBP, but Have no idea how to start and prove... need help
1
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2answers
27 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 ...
2
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0answers
66 views

How do I to solve this special type of integral

Does the following function can be simplified or solved? $$R(i) = \int_{y\in S} {\frac{{w(y) g(y,i)_{}^\sigma }}{{\int_{x\in S} {h(x)g(x,y)_{}^\sigma f(x,y)_{}^\sigma dx} }}dy} $$ where S is a ...
1
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0answers
23 views

Integral of action (quantum field theory, prescription)

I am struggling to show: $$\int_{w=0}^{w} \int_{r=2M}^{2(M-w)} \frac{-drdw}{1-\sqrt{\frac{2(M-w)}{r}-\frac{Q^2}{r^2}}}=2\pi[{2w(M-\frac{w}{2})-(M-w)\sqrt{(M-w)^2-Q^2)}+M\sqrt{M^2-Q^2}}]\\$$ with ...
0
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0answers
37 views

Finding the integral of a 1/variable*radical function

I'm trying to find the integral of $$\int\frac{1}{x* (\sqrt{4x^4 - 9})}$$ Attempt: I assumed that the integral would be some sort of inverse trigonometric function. Because of this, I did the ...
1
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2answers
72 views

Calculate $\int_{0}^{1}\left \{ \sqrt{1-x^2}+2 \right \}^2 dx$

I couldn't find any suitable substitution for this integral and hence I couldn't solve it. $$\int_{0}^{1}\left \{ \sqrt{1-x^2}+2 \right \}^2 dx$$
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0answers
20 views

Integral of combination of power, exponential, and kummer hypergeometric function

I am trying to solve a couple integrals of the form: \begin{equation} \int_{0}^{\infty} x \, e^{-a(x-b)^{2}}\, M(-\alpha,-\beta,\lambda x) \end{equation} $\alpha > 0$ and $\beta > 0$ are ...
0
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2answers
52 views

Evaluate the definite integral $\int_{0}^{a}\frac{dx}{(a^2+x^2)^{3/2}}$

I'm trying to solve this integral with trigonometric substitution but am having a ton of trouble: $$\int\limits_{0}^{a}{\frac{dx}{(a^2+x^2)^{\frac{3}{2}}}}$$ I tried $x=a\tan{\theta}$ and thus ...
0
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2answers
36 views

What is the value of integral? [on hold]

Let $y(t)$ be a continuous function on $[0,\infty)$. If $$ y(t)= t\left(1-4 \int^t_0 y(x) dx\right) +4 \int^t_0 xy(x) dx$$ then what is the value of $\int^{\frac{\pi}{2}}_0 y(t) dt\,$?
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1answer
42 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... ...
1
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1answer
35 views

Graphical Convolution

For the first part of the above problem, I copied an example from my book and I got the answer to be $$t(t-1)+t(t-2)=t^2-3t$$ considering that the integral is the sum of the area of the rectangles ...
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0answers
24 views

Evaluate $\int_{-\infty}^{\infty}\sin(k_0 \xi)e^{-\frac{(x-\xi)^2}{4a^2 t}}d\xi$

I have a heat equation for which in the solution I need to evaluate the following integral: $$\int_{-\infty}^{\infty}\sin(k_0 \xi)e^{-\frac{(x-\xi)^2}{4a^2 t}}d\xi$$ Except of the common gaussian ...
0
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1answer
26 views

Parametrize the given curve and compute the integral (complex numbers)

The integral I have to evaluate is $\int_Czdz$, where $C$ is the line from 0 to $1+i$, and then from $1+i$ to 2. My work: $z_1(t)=(1+i)t$ and $z_2(t)=(t+1)+i(1-t)=t(i-1)+(1+i)$, $t\in[0,1]$. ...
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0answers
31 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
13 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 ...
1
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2answers
27 views

how could calculate $ \int_{C} \frac{1}{\sin(z)} \, dz $ when $C=C(0,1)$

i am trying calculate $$ \int_{C} \frac{1}{\sin(z)} \, dz $$ when $C=C(0,1)$ by complex methods, its said, by residues, some one could help me?
1
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0answers
17 views

Volumes by Cylindrical Shells - What am I doing wrong?

I am trying to solve this exercise from a textbook: $y = x^4, y = 0, x = 1;$ rotated about $x=2$ This is my attempt at solving the problem: Shell radius: $2 - x$ Shell height: $x^4$ $a = 1$ $b = 2$ ...
0
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0answers
74 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 ...
2
votes
2answers
26 views

How could I calculate $\int_{C} ze^{\frac{1}{z-1}}$ when $C=C(1,\frac{1}{2})$

I have to solve if $C=C(1,\frac{1}{2})$ $$I=\int_{C} ze^{\frac{1}{z-1}}$$ I know that $I=2\pi i \operatorname{Res}(f(z), 1)$, but I do not know how could I calculate that residue. What I did: ...
0
votes
1answer
39 views

Infinite Sum Defined by $\int \frac{e^x}{x}dx$ vs. Exponential Function Taylor Series

Recently, when fiddling around with integration by parts, I noticed that it is possible to define infinite series that led to an integral. My calculus teacher noticed this, and told me to find $$ ...
0
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0answers
14 views

Determining the unit normal field of a paraboloid $P$, and integrating a vector field over $P$

Let $M \subseteq \mathbb{R}^n$ be a $n-1$-dimensional manifold, and $N_x M$ the normal vector space of $M$ at a point $x \mathbb{R}^n$, that is, the (1-dimensional) space of vectors that are ...