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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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 $t=rsinx$) So, immediately I did $dt=rcosx\,dx$ and substitute it all in... $\int ...
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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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13 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 ...
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13 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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23 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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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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2answers
20 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?
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12 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: $x^4$ $a = 1$ $b = 2$ ...
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60 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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2answers
24 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: ...
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34 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 $$ ...
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7 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 ...
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13 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 ...
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44 views

find the area of a kite with integration

A stunt kite has the shape in the diagram below: How can I find the area using calculus integration. Can anyone help me start this question, I am not looking for the full answer. I assume I only ...
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1answer
13 views

Surface Integral of $3z^2 d\sigma$

Let $S$ be the bounded surface of the cylinder $x^2+y^2=1$ cut by the planes $z=0$ and $z=1+x$ Then how to show that the value of the surface integral $∬3z^2 d \sigma $ over $S$ is equal to ...
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52 views

Conjecture $\int_0^{1}\frac{{(\rm arcsin})^2({x^2})}{\sqrt{1-x^2}}dx\stackrel?=\frac{5}{24}{\pi^3}…$

$$I=\int_0^{1}\frac{{(\rm arcsin})^2({x^2})}{\sqrt{1-x^2}}dx\stackrel?=\frac{5}{24}{\pi^3}-\frac{\pi}{2}ln^2{2}-2{\pi}\operatorname\chi_{2}(\frac{1}{\sqrt{2}})$$ This result seems to me digitally ...
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38 views

Integration of $\frac{x^2}{2\left(e^x+1\right)}$

Let: $$f(x) = \int \frac{x^2}{2\left(e^x+1\right)}dx $$ Is there a way to find $f(x)$? I've tried through integration by parts, but that didn't work out. If substitution is the answer, I can't see ...
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1answer
56 views

Is there a nice closed form to $\int_0^{\pi/2} (\log \sin x)^n\text{ d}x$ for $n\in \Bbb{Q},n\gt 1$?

For $n\in \Bbb{N}$, $$\int_0^{\pi/2} (\log \sin x)^n\text{ d}x=\frac{1}{2^{n+1}}B^{(n)}\left(\frac{1}{2},\frac{1}{2}\right)$$ Can we extend that result a bit further, to $n\in \Bbb{Q},n\gt 1?$
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5answers
67 views

integration of $1/x$ a counterexample to the rule

We know that the integration of $\displaystyle\int\frac{1}{x}\,dx=\log\left(|x|\right)$+$c$ with $x\neq 0$ , but if we go by normal rule then it becomes $\infty$. Is this a counterexample to the rule ...
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43 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!!
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3answers
43 views

what's wrong with this integral by parts calculation?

Today, I just finda confusing quesiton. If I do this way: $$\int { \frac { dx }{ x } } =x\left( \frac { 1 }{ x } \right) -\int { xd } \left( \frac { 1 }{ x } \right) =1+\int { \frac { dx }{ x } } ...
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1answer
17 views

Examples of physical motivation for integrals over scalar field?

I'm looking for good examples of physical motivation for integrals over scalar field. Here is an example I've found (source): A rescue team follows a path in a danger area where for each position ...
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43 views

How to integrate the following function (somewhat similar to Thomaes function)

We have not covered and thus it is not valid to use ideas such as Lebesgue integration, measure, etc. I was given a hint to use either squeeze theorem, or the criterion about if Riemann integrable ...
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1answer
38 views

Splitting integral !!

i have this simple question that make me really confused : let $\phi$ a smooth function with compact support and $p>0,t\ge 0$ and : \begin{equation} U_p= \left\lbrace \begin{array}{ccc} 0 & ...
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1answer
27 views

parametric equations, finding the range of t

When parametrizing a curve how doe we obtain the range of $t$? For example lets say we have the parametrization: $x(t) = 1+3t$ and $y(t) = 2+5t$. How do we find the range of t? $t\to[?,?]$
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19 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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21 views

Independence of parametrization in defining integral of differential form

This is an exercise from Spivak's Calculus on Manifolds. Questions asks the following : (Independence of parametrization). Let $c$ be a singular $k$ cube and $p:[0,1]^k\rightarrow [0,1]^k$ be ...
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22 views

Does the quadratic covariation process define a measure?

In the context of stochastic integration (when we define the space $L^2(M)$), we define the (possibly infinite) measure $$P_M = P \otimes [M]$$ by $$E_M[Y] = E\left[\int_0^\infty Y_s(\omega) ...
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Conjecture: $\int_0^{\infty}dx\frac{e^{i\alpha\sqrt{x^2+1}}}{\sqrt{x^2+1}}J_1(Qx)=\left(e^{i\alpha}-e^{i\sqrt{{\alpha}^2-Q^2}}\right)/Q$

Here $\alpha>0$, $Q>0$, and $J_1$ is a Bessel function. I'm fairly certain the closed form in the title is accurate for a couple of reasons. First, I've evaluated the integral numerically in ...
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2answers
37 views

ELI5: Riemann-integrable vs Lebesgue-integrable

I am wondering what the difference between riemann-integrable and lebesgue-integrable means. Does it have anything to do with the absolute value of the integrand, something like ...
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1answer
28 views

$E=\left\{(x_1,x_2,x_3,x_4):\sqrt{x_2^2+x_3^2+x_4^2}\le x_1\right\}$, what is $\int_E e^{-\langle x,t\rangle} \, dx$?

I'm learning some real analysis and encountered the following question: Let $E=\left\{(x_1,x_2,x_3,x_4)\in \mathbb{R}^4:\sqrt{x_2^2+x_3^2+x_4^2}\le x_1\right\}$. for which values of ...
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2answers
18 views

Rotational Volume

I have to find the volume of the region bounded by $ y= \sqrt{x-1} $, y=3, the y-axis and the x-axis rotated around y=5 I set up $\int_1^{10} $ $\pi((5-(\sqrt{x-1}))^2 - (5-3)^2)$dx + $\int_{0}^1$ ...
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1answer
65 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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20 views

integration of product of error function

Is there a more simple formulae for the following integral $$ \int_{a}^{+\infty} erf(\alpha x).erf(\beta x) \frac{1}{x^2} \:\mathrm{d}x $$ where $a>0$, $\beta>0$ and $\alpha>0$
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1answer
46 views

Show $ (\int_{-\infty}^\infty \sqrt{p}\sqrt{q}d\mu)^2\leq 2 \int_{-\infty}^\infty \min\{p,q\}d\mu $

Consider a random variable $X$ in $(\Omega, \mathcal{F}, \mathbb{P})$. Let $p,q$ be two densities with respect to a measure $\mu$ in $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ where ...
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2answers
102 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? ...
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2answers
45 views

How to get this result of integral?

Statement \begin{equation} \int_{\mathbb R} \exp \left( -2\pi (\frac{x}{\sqrt{2}})^2 \right) \exp\left( -i2 \pi \frac {x}{\sqrt{2}} \cdot f \right) dx = \exp \left( -\pi f^{2} / 2 \right) ...
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48 views

Sine improper integral

Suppose the following integral $$ \int\limits_{-\infty}^{\infty}\sin{x}dx $$ In mathematical rigor, the following is the definition $$ \int\limits_{-\infty}^{\infty}\sin{x}dx = ...
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4answers
58 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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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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Prove that a function is Riemann integrable directly, using $\epsilon-P$

I know there already are questions like these, but I still don't understand how to prove it. Question: Prove that $f$ is Riemann integrable on $[0,1]$ if $$f(x) = \begin{cases} x^2 \sin (1/x) ...
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3answers
23 views

Substitution and Partial Fractions (Integration)

$$\int\frac{dx}{x-\sqrt[4]{x}}$$ given the substitution $x=u^{4}, dx=4u^{3}du$ $$=\int\frac{4u^{3}du}{u^{4}-u}=\int\frac{4u^{3}du}{u(u^{3}-1)}=\int\frac{4u^{2}du}{(u^{3}-1)}$$ At this point I ...
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1answer
48 views

How do you differentiate the integral from $ \int_{e^{-x}}^{e^x} \sqrt{1+t^2}\,dt$ [duplicate]

How do you differentiate the integral from $e^{-x}$ to $e^x$ of $\sqrt(1+t^2)$ with respect to t? $$ \int_{e^{-x}}^{e^x} \sqrt{1+t^2}\,dt $$ I know the answer is $$ e^x\sqrt{1+e^{2x}} + ...
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1answer
56 views

To refute : a function with one discontinuity point is integrable on $\left[0, 1\right]$

If, let $f: \left[ 0, 1\right] \to \mathbb{R}$ continuous with only one discontinuity point is integrable on $\left[ 0, 1 \right]$. I think this is false but I can't find a example that contradicts ...
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1answer
34 views

How to calculate the integral of exponential of complex exponential?

How to express $$\int^{\pi}_{-\pi}e^{c \cdot e^{j\omega}}d\omega$$ in closed form, where $c$ is a constant? Should it be some Bessel function?
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1answer
25 views

References for the “extended” Green and Stokes' theorem.

I was watching these videos from MIT's series: Green, Stokes. And I didn't understand the justification: their "extended version" of the theorems. I looked up on google and couldn't find many ...
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20 views

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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1answer
16 views

Error Bounds with Trapezoidal Formula

I know there are some posts about the same thing but I am unable to do my specific question or at least, I don't think I'm doing it the right way. The question asks me to use the Trapezoidal Error ...
3
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79 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
73 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 $$I=\int_0^{1}\frac{{ \arcsin}({x^2})}{\sqrt{1-x^2}}dx$$