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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1answer
43 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 ...
1
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1answer
47 views

To prove or refute: $\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^{N} f\left( \frac{n}{N} \right) = 1$ then $f \in R\left( \left[ 0, 1 \right] \right)$ [on hold]

Let $f : \left[ 0, 1 \right] \to \mathbb{R}$ such that $$\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^{N} f\left( \frac{n}{N} \right) = 1.$$ Then, $f \in R\left( \left[ 0, 1 \right] \right)$ and ...
2
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1answer
56 views

Integral of $\int_0^{2\pi}{e^{({x\sin\theta)}}\ \text{d}\theta}$

I would like to compute $$f(x)= \int_0^{2\pi}{e^{({x\sin\theta)}}\ \text{d}\theta}$$ I already know that it equals $2\pi\sum_{j\geq0}{(\frac{x^j}{2^jj!})^2}$. I'm already happy since it provides an ...
-2
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1answer
31 views

Evaluate a double integral over a region $R$

Let $R$ be the refion enclosed by $x^2+4y^2\ge 1$ and $x^2+y^2\le 1$. Calculate $$\iint_R \lvert xy\rvert\,dxdy$$ I think the answer is $0$ because the area in the positive quadrants cuts the ...
2
votes
3answers
71 views

Dealing with integrals of the form $\int{e^x(f(x)+f'(x))}dx$

Integrals of the form $$\int{e^x(f(x)+f'(x))}dx$$ are very common. And I have seen this form appearing in several exam papers.But the problem I face with this particular type of integral is finding ...
3
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1answer
93 views

integrate $\int \frac{dx}{4+3sin2x}$

$$\int \frac{dx}{4+3\sin (2x)}$$ $u=2x$ $du=2dx$ $$\frac{1}{2}\int \frac{du}{4+3\sin(u)}$$ $v=\tan(\frac{u}{2})$ $du=\frac{2dv}{1+v^2}$ \begin{align*} \frac{1}{2}\int ...
0
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0answers
25 views

Determination of a formula of an area of shape limited by curves using integrals?

I did two examples of determining an area limited by curves and I'm not sure if I did them right. I would really appreciate if someone checked my solution and fixed any possible mistakes. Ex.1 Area ...
1
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0answers
27 views

Discontinuous parametric integral function

Is there an example of a function $f:[0,1] \times [0,1] \to \mathbb{R}$ such that for all $x \in [0,1]$ the function $\phi(y) = f(x,y)$ is continuous in $y$ and for all $y \in [0,1]$ the function ...
1
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1answer
29 views

Calculation of area in 2 definite integrals given function $y=x^2$

Here is a graph for $y=x^2$ Given that the area in blue is equal to the area in pink, find a in terms of b and solve for a. My attempt: From the graph I can see that :$a^2=b$ and $a=\sqrt b$ ...
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1answer
19 views

Finding Intervals after Changing the Order of Integration

Problem: Let $f(x) = \int_0^x e^{t^2} \,dt.$ Find the average value of f on the interval $[0, 1]$. Thoughts: $\int_0^x e^{t^2} \,dt$ is a non-elementary integral. The average value of f, I believe, ...
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0answers
20 views

The line integral of the polar angles of points of the XY plane through a closed curve

Let me ask the following question. Let $XY\setminus \{(0,0)\}$ denote the 2D XY plane excluding the origin point. And let $\mathbb{R}$ be the set of all real numbers. Let a function $f:XY\setminus ...
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0answers
75 views

If $y(t) = t\left(1-\int_0^ty(x)\,dx\right)+4\int_0^tx\,y(x)\,dx,$ then $\int_0^{\pi/2} y(t)\,dt$ is equal to?

Leibniz rule or Laplace transform? Let $y(t)$ be a continuous function on $[0,\infty)$. If $$y(t) = t\left(1-4\int_0^ty(x)\,dx\right)+4\int_0^tx\,y(x)\,dx,$$ then $\int_0^{\pi/2} y(t)\,dt$ ...
0
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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 ...
3
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2answers
83 views

How to solve integrals where you can't factor a polynomial?

Hi there guys I don't know if the title of the question should be the one for this but the thing is that I'm trying to solve this integral $\int \frac {\frac 12-u^2}{2u^4-2u^2+1}$$du$ and I have this ...
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1answer
74 views

How to solve this definite integration [on hold]

$$ I = \int\limits_0^\pi \frac{d\theta}{\left[(\alpha - \beta \cos \theta)^2 + c \right]^2 + d^2} $$ Source. I'm looking for a numerical method/scheme which can be used to solve the following ...
4
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3answers
284 views

Why is the surface area of a sphere not given by this formula?

If we consider the equation of a circle: $$x^2+y^2=R^2$$ Then I propose that the volume of half of a sphere of radius $R$ is given by the summation of the circumferences of the circles between the ...
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3answers
48 views

How to calculate $f(x, y, z)$ given $d f = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy +\frac{\partial f}{\partial z} dz$

On a manifold with local coordinates $(x_1, \ldots, x_n)$ I have a closed 1-form $\omega$ for which $d \omega = 0$ holds. This means There must be a function $f(x_1, \ldots x_n)$ for which $d f = ...
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2answers
72 views

How to compute $\int_0^2(1+4t^2+9t^4)^{1/2}\text{d}t$?

The original question was: find the length $\ell$ of the curve $\gamma$ given the parametric equations: $$x=t~~~~~ y=t^2~~~~~ z=t^3 $$ from $t=0$ to $t=2$
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0answers
30 views

Laplace Transform of $e^{t^3}$

I have to find the Laplace transform of $$e^{t^3} u(t)$$ and I know that $u(t)$ will just change the integral from negative infinity to positive infinity to $0$ to positive infinity, but I'm stuck ...
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0answers
16 views

Integral of dot product of unit vector

I am having trouble with the following integral. $$\int \left(\bar{A} \cdot \hat{ F\left(\lambda\right)}\right)^p\mathrm ds$$ Note that the right hand side of the dot product is normalised. Where: ...
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2answers
27 views

Constancy of an integral function

Fix some $\ell\in\mathbb{R}^+$. Say that $f:\mathbb{R}^2\to\mathbb{R}_{\geq0}$ and $\mu:\mathbb{R}\to\mathbb{R}^+$ are functions satisfying the following: $f$ and $\mu$ are continuous. $f$ is ...
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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
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1answer
11 views

Volume Exponential Function

I should find the Volume received by rotating the region bounded by: $y = e^x $, $ y = 0 $,$ x = 0 $, $ x = 1 $ rotated around the x axis. I know how to find it by using the disc method but I could ...
1
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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 ...
2
votes
3answers
79 views

integrate $\int\frac{\sin x}{1+\sin^{2}x}dx$

$$\int\frac{\sin x}{1+\sin^{2}x}\mathrm {dx}$$ $$\int\frac{\sin x}{1+\sin^{2}x}\mathrm {dx}=\int\frac{\sin x}{2-\cos^{2}x}\mathrm {dx}$$ $u=\cos x$ $du=-\sin x dx$ $$-\int\frac{\mathrm ...
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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 ...
1
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1answer
25 views

Example of a non square-integrable martingale?

Are there (simple) examples of martingales which aren't square integrable?
0
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1answer
28 views

use L1-convergence to show integral convergence

Let $f\in L^1([0,1])$, $g_n$ a sequence of continuous functions that converges in $L^1$ to some $g\in L^1([0,1])$. Now my question is: Does $\int_0^1 f(t)e^{g_n(t)} dt$ converge to $\int_0^1 ...
2
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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 ...
3
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1answer
44 views

Which inequalities are there with stochastic integration?

Which inequalities can I use with stochastic integration? For example, with the standard lebesgue integral we have $$\left|\int_\Omega f(x) dx\right| \le M |\Omega|$$ (where $M$ is the maximum of ...
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1answer
29 views

Changing the order of a double integration ? $\int_{-5}^{5}dx\int_{-7}^{\sqrt{25-x^2}}f(x,y)dy$

I've been doing an example of changing the order of a double integral and I'm not sure if I did it right. I would really appreciate if someone would check if my solution is right and correct any ...
1
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1answer
39 views

How to show that this is a martingale?

Let $H_s$ be a predictable and bounded process. How can I show that $$M_t = \int_0^t H_s \, dW_s$$ is a martingale? Clearly since $H_s \in L^2_\text{loc} (W)$ we have that $M_t$ is a local ...
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0answers
24 views

To what Sobolev space does this function belong to?

I am given this function: $$f(x) = e^{- \sqrt{|x|}}$$ and I want to find $k\in \mathbb{N}, \ p \ge 1$ such that $f \in W^{kp} (\mathbb{R})= \{ f \in L^p (\mathbb{R}) \ | \ \forall \alpha \le k: \ ...
0
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1answer
26 views

Integrating this norm

The norm \begin{equation} ||u||^{2} = \int_{\mathbb R} 1 \cdot u(x) \cdot \overline{u(x)} dx \end{equation} Claim which should be correct \begin{equation} ||u||^{2} = \int_{\mathbb R} 1 \cdot u(x) ...
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} ...
1
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2answers
60 views

Integrating triangle in a 2D plane

I am interested in integrating $(x^2y+y^2x)$ on the following loop: $(x=1,y=2)\rightarrow(x=2,y=1)\rightarrow(x=3,y=3)\rightarrow(x=1,y=2)$. I know this loop forms a triangle with all three sides ...
1
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1answer
26 views

When should we use absolute value by solving this integral?

I have for example: $\int_{y_0}^{y} \frac{1}{2\eta} d\eta$ with $t_0$ a real constant and the solution was: $\frac{1}{2}(ln|y|-ln|y_0|)$ But on other case I had: $\int_{y_0}^{y} \frac{1}{\eta} ...
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1answer
21 views

Complex integral over line, similarity with conservative field

Have $\int_C(4z^2-2iz)dz$ integral. Does it depend on choice of path? Tried to express $f(z)=(4z^2-2iz)$, then $f(x+yi)=(4x^2-4y^2+2y)+i(8xy-2x)$ Then $\frac{\delta P}{\delta y}=-8y+2$ And ...
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0answers
28 views

Does this function exist an inverse function?

Could I find the inverse function of the following integral equation? I am going to write it as $h(x)=...$ The integral equation is: $$R(i) = \int_{y\in S} {\frac{{w(y) g(y,i)_{}^\sigma ...
4
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2answers
78 views

Find the integral $\int \frac{1+x}{\sqrt{1-x^2}}\mathrm dx$

The integral can be represented as $$ \int \frac{1+x}{\sqrt{1-x^2}}\mathrm dx= \int \left(\frac{1+x}{1-x}\right)^{1/2}\mathrm dx $$ Substitution $$t=\frac{1+x}{1-x}\Rightarrow ...
0
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1answer
25 views

Finding the center of mass for a centroid without a convenient symmetry axis

Find the centroid of the lamina described in polar coordinates as $\left \{ \strut \left ( x,y \right )~|~0\leq r\leq 4 \cos\left ( \theta \right ),0\leq \theta \leq \frac{\pi}{3} \right \}$ Having ...
2
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1answer
20 views

Calculate the surface integral of bounded cylinder

Evaluate $$\int\int z^2\,dS,$$ where $S$ is the part of outer surface of cylinder $x^2+y^2=4$ between the planes $z=0$ and $z=3$. The answer given in book is $\pi$ but I am not getting this ...
1
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1answer
34 views

Riemann integrability given by limit of $\frac 1n \sum_{k=1}^n[f(k/n)]$

If $f:[a,b] \to \mathbb R$ is such that $$\lim_{n \to \infty} \frac 1n \sum_{k=1}^n[f(k/n)] = 1,$$ does that imply that $f$ is Riemann integrable on $[a,b]$ ? Thank you.
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0answers
19 views

I need some referenced materials about such type of integral.

I am struggling with the following type of integral. I can't find any referenced materials of it. Could you recommend some for me? Any books, papers or lecture notes are ok. Thank you. $$R(i) = ...
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2answers
31 views

Find the area bounded by function $y^2=16-2x$, the tangent to the curve at the point $(6,2)$ and the $y$-axis

First we find a tangent line of function $y^2=16-2x$ at $T(6,2)$: $y_t-y_0=f'(x_0)(x-x_0)$ where $x_0=6,y_0=2$ $y^2=16-2x\Rightarrow y=\sqrt{16-2x}$ or $y=-\sqrt{16-2x}$ Derivative of ...
3
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0answers
38 views

Why would an equation switch signs when something becomes independent of time? (Traffic Flow)

EDIT: I'm too tired for math and the answer to my question is explained in a comment below. Should this post be removed? Not sure if it adds much to the community given that it was all just me ...
0
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0answers
28 views

Baby Rudin theorem 6.16, explanation that a Riemann Stieltjes integral could be expressed as a infinite series.

The theorem says: Suppose $c_n \geq 0$ for $1,2,3 ...$. $\sum c_n$ converges, $\{s_n\}$ is a sequence of a distinct points in $(a,b)$, and $\alpha (x) = \sum^{\infty}_{n=1} c_n I(x-s_n)$. Let $f$ be ...
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0answers
41 views

I have a question about calculating integral

I need some help to solve this integral: $$\int_{4x}^{\infty} \frac{w^{\frac{m}{2}}e^{-a\sqrt{w+2\sqrt{xw}}}}{\sqrt{w^2-4xw}}\mathrm{d}w.$$ Thank you
0
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0answers
8 views

Gaussian weighted intergal of Product of Gaussians

I'm trying to find a solution to the following function, My understanding is that the resultant function should still be a Gaussian, however I would like to define it as a linear function the ...
1
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2answers
31 views

Finding distance of $h(t)=t$ from a closed subspace $Y$ of $\pi$-periodic functions in $L^2(-\pi,\pi)$

Let $Y=\{f\in L^2(-\pi,\pi):f(t-\pi)=f(t) \text{for almost all $t\in(0,\pi)$} \}$ Show that there exists $g\in Y$ such that $$\|h-g\|_2=\inf \{\|h-f\|_2:f\in Y\}$$ where $h(t)=t$. Compute ...