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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15 views

under which conditions this equality holds

Consider $f : [0,\infty) \rightarrow \mathbb{R}$ be a function such that $\lim_{t\rightarrow \infty} f(t) = 0$. I was wondering if the following relation holds $$lim_{t\rightarrow\infty}\int_0^t ...
0
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1answer
46 views

Integral of $xe^{-ax^2-bx^{-1}}$

I am currently facing an integral I have no clue how to solve it. I believe it is rather exoctic, but I hope you might have some good advice: $$\int_0^{\infty} x e^{-ax^2-bx^{-1}} \, \mathrm{d}x, ...
1
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0answers
22 views

integral inequalities and continuous functions

Let $f$ be a positive, continuous function on $\mathbb{R}$. Let $c\in (0,1/2)$ be a constant and $\lambda>1$. I want to prove that: (1). for any $a\in\mathbb{R}$, there exists $\delta(a)>0$ ...
0
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1answer
12 views

Find the density function of $X$, from the random vector $(X,Y)$ if the PDF of this vector is:

$$\phi(x,y)= \frac{|x|}{\sqrt{8 \pi}}e^{-|x|- \frac{1}{2}x^2y^2}, x,y \in R $$ Now I'm aware I would have to do $$\phi_X(x)=\int_{- \infty}^{+ \infty}\phi(x,y) dy$$, what is confusing me is this ...
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0answers
30 views

How to integrate $\int dx \frac{1}{\cosh^2 x +a^2}$ [on hold]

How to integrate: $$\int dx \frac{1}{\cosh^2 x +a^2}$$
3
votes
2answers
48 views

How to prove $\lim \limits_{n\to\infty}(n+1)\int_{0}^{1}x^nf(x)dx=f(1)$

I need help to prove this in real analysis. I think it uses IMVT, but not sure how to do it. Let $f(x)$ be a real valued continuous function on $[0,1]$. Show that $$ \lim ...
2
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0answers
22 views

Finding area of a spheroid

Let $M=\{(x,y,z)\in \Bbb{R}^3 : (x/a)^2 + (y/b)^2 + (z/c)^2 = 1\}$. Find $\text{vol}_2(M) = \int_M 1 dS$. My attempt: The map $$\Phi:(0,\pi)\times (0,2\pi)\to \Bbb{R}^3\\ \qquad (\varphi, ...
0
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2answers
52 views

Integral of $e^{x^3}$

How do I find the integral of $e^{x^3}$. I have to do find the following integral and when I try to do integration by parts, I cannot find the integral of $e^{x^3}$. $$\int x^2 e^{x^3} ...
0
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1answer
27 views

Calculating volume by disc integration

What is the volume $V$ of the object created when the area formed by the lines $$y=x$$ $$y = 2-x^2$$ $$0 \le y \le 2$$ is rotated around the $y$-axis? It says that the answer is $\dfrac{5\pi}{6}$. ...
0
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2answers
25 views

Surface area of revolution of curve

I am wondering why this particular integration is being found difficult to solve. Would appreciate any help I can get. the graph is $y = x^3$ and the limits are $0 \leq y \leq 1$
2
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1answer
69 views

$1-1+1-1+1-1+\cdots = \frac 12$ proof?

Note: I claim no credit for this "proof". A friend came up with it and I thought it was pretty cool. Let's say you wanted to prove that $$1-1+1-1+1-1+ \cdots = \frac 12$$ Well, as mentioned before, ...
0
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0answers
12 views

Convergence in distribution of distributions $p_n$ implies convergence in distribution of $s_n$?

Question Setup Suppose $p_n(x,y)$ is a sequence of probability densities on $\mathbb R^2$ and $q_n(x)$ is a sequence of densities on $\mathbb R$ such that \begin{align*} \int b(x,y) \ p_n(x,y) \ dx ...
0
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1answer
21 views

Measurable function growing at most linearly

Let $F$ be a measurable function on $\mathbb{R}$ which grows at most linearly ($F(x) \leq C|x|$), and is differentiable at zero, $F'(0)=a$. Show that $$\lim_{n\rightarrow \infty}\int_{-\infty}^\infty ...
4
votes
1answer
44 views

Improper Intergral (Fresnel- like)

Let $\alpha >1$. Show that $$\int_0^\infty \sin(x^\alpha)\,dx= \sin\left(\frac{\pi}{2\alpha}\right) \int_0^\infty e^{-r^\alpha}\,dr.$$ I was going to ask how to do this but figured it out while ...
0
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0answers
14 views

Understanding Verlet Velocity Method

How does the Velocity Verlet method differ from the standard Euler method? Why do we need to add Acceleration / 2 to calculate position?
0
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0answers
16 views

Divergence theorem for a second order tensor

I want to integrate by part the following integral in cylindrical coordinates $$\int \vec{r} \times (\nabla \cdot \overline{T}) ~d^3\vec{r} $$ where $\overline{T}$ is a second order symmetric tensor ...
-2
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1answer
43 views

How to integrate a function with a nested absolute value: $|x^2 - 2|x||$? [on hold]

I need help with this problem, $$\int_0^4|x^2 - 2|x||dx$$ what should I do with $2|x|$ ?
0
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0answers
18 views

Cylindrical Coordinates

What is the following integral in cylindrical coordinates? I did it like this and I get pi/5, but if I solve in cartesian I get 2/5.
1
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2answers
51 views

Expectation of $\mathbb{E}(X^{k+1})$

I have difficulties with an old exam problem : Let $X$ be a positive random variable defined on a probability space $(\Omega, \mathcal{F}, \mathbf{P})$. Show that $$\int_0^\infty t^k ...
0
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0answers
11 views

Calculate rotational volumes

I need to calculate the volume from rotating f(x) around y=2x using Pappus–Guldinus theorem. For that I need to know the distance A. $$L = (f(x) - 2x) / 2$$ But how can I optain the distance A?
1
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1answer
25 views

Does this kind of integral rearrangement work?

I know there isn't any general closed form for $\int{1\over{f(x)}}dx$, but let's say that for some function g(x) the following anti-derivative holds: $$\int{1\over{f(x)}} dx={g(x)\over{f(x)}}$$ Does ...
0
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0answers
8 views

Optimal Space-Travel Departure Time (Issues deriving and solving complex expressions).

Problem This problem aims to determine the optimal time to depart for an intergalactic destination, taking into account the fact that in a number of years technology back on the planet you left may ...
2
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0answers
24 views

upper bound of a differential equation solution

Let $A(t)$ be a bounded singular values matrix that is function of time, and $f(t)$ and $L^\infty$ function of time. And consider the ODE $$ \dot x = A(t) x + f(t) $$ How we can describe qualitatively ...
11
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2answers
95 views

Why is $\int\int f(x)f(y) |x-y|dxdy$ negative?

The Setup Let $f:\mathbb{R} \to\mathbb{R}$ be a smooth function with support in the interval $[-R,R]$ and satisfying $\int f = 0$. By manipulating some integrals, I found the surprising inequality ...
0
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0answers
31 views

upper bound of an $L^\infty$ function's derivative

Consider a function $u:\mathbb{R} \longrightarrow \mathbb{R}^n$ that is essentially bounded, i.e., $u \in L^\infty$. There is an upper bound of its derivative? I think there is not allways ( i.g. ...
1
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1answer
31 views

complex integral of 1/z independent of choice of ellipse?

Can Someone please help me with the following. complex integral of 1/z over an ellipse is independent of choice of ellipse centered at zero. Why is this the case. Is it due to homotopy ...
0
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1answer
56 views

How to integrate $e^{-t^2}$? [duplicate]

Anyone know how to integrate the following? $$ \int_0^{+\infty} \! e^{-t^2} \, \mathrm{d}t $$ Thanks
0
votes
0answers
23 views

Stieltjes integral with discontinuous integrator

I am asked to solve the following Stieltjes integral: Compute $\int_0^6 f\, dg$, where $f(x) = 6x-x^2$ and $g(x)$ is defined by: $$ g(x) = \left\{ \begin{array}{ll} x^2 &\hbox{for $0\leq x ...
0
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2answers
40 views

Expanding the integrand

Can anyone help me find the solution to this integral: $$\int\limits{(t-4)(t-2)^{4/5}}dt?$$ I think I need to expand the integrand but I do not know how. Thanks a lot!
4
votes
3answers
285 views

Decomposition into partial fractions to compute an integral

I'm having problems with: $$\int_{-\infty}^{\infty}\frac{x^4+1}{x^6+1}dx$$ I was thinking: $\frac{x^4+1}{x^6+1}$ is an even function and the interval $(-\infty,\infty)$ is symmetric about 0, we ...
1
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0answers
21 views

Covariance of two integrated Brownian motions

I have a question that is similar to the one here: covariance of integral of Brownian, but the answer that I come up with does not match what the book claims the answer is. Given that $$X_t = ...
0
votes
3answers
52 views

Calculate $\lim_{n\rightarrow \infty}\int_{[0,1]}\frac{n\cos(nx)}{1+n^2 x^{\frac{3}{2}}}$

I have tried several methods but even I can not calculate. $$\lim_{n\rightarrow \infty}\int_{[0,1]}\frac{n\cos(nx)}{1+n^2 x^{\frac{3}{2}}}\,dx$$ If anyone can help, it was part of a test and still I ...
1
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0answers
28 views

Help With Limit of Integral

So I am working through some practice problems, and on one of them I can't get the second part: For $x\in(0,\infty)$ and $n\in\{1,2,3,\dots\},$ let $f_n(x)=\frac{e^{\sin\left({x^2/n}\right)}}{1+x}.$ ...
0
votes
1answer
12 views

Revolving an unknown equation around the x and y axes

The first quadrant region enclosed by the x-axis and the graph of y = ax - x^2 traces out a solid of the same volume whether it is rotated about the x-axis or the y-axis. What is the value of a?
0
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2answers
78 views

Suppose $a<b<c<d$ and $p(x)=(x-a)(x-b)(x-c)(x-d)$. Show that $\int_a^b \frac{dx}{\sqrt{|p(x)|}} = \int_c^d \frac{dx}{\sqrt{|p(x)|}}$

Suppose $a<b<c<d$ and $p(x)=(x-a)(x-b)(x-c)(x-d)$. Show that $$\int_a^b \frac{dx}{\sqrt{|p(x)|}} = \int_c^d \frac{dx}{\sqrt{|p(x)|}}.$$ My attempt: I perform linear substitution $u=x-a+c$ ...
0
votes
1answer
36 views

What can be said about $f''$ if the trapezoidal approximation is always an overestimate?

For any $a$ and $b$ the Trapezoidal approximation of the integral $\int_a^b f(x)\,dx$ is an overestimate. What can you conclude about the second derivative of $f$? I think it might mean that the ...
2
votes
3answers
65 views

Real Methods to Evaluate $2 \int_{-1}^{1}x^2 \sqrt{1-x^2}dx$

I was recently contacted by a friend to find the values of the two following integrals by any means. $$ I=2\int_{-1}^{1}x^2 \sqrt{1-x^2}dx$$ $$ J=\int_{-1}^{1}(1-x^2) \sqrt{1-x^2}dx$$ The first ...
6
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1answer
85 views
0
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0answers
22 views

Bayesian statistics and Basis for continous functions

I was thinking about Bayesian statistics, and one thought bothered me: In Bayesian statistics, we assume that the pdf $p(x)$ can be described as: $p(x)=\int f(x|\theta)g(\theta)d\theta$ usually ...
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0answers
39 views

What is the integral of $ \int \frac{-6000} { (3x+50)^2} dx$ [on hold]

How can I find the value of the integral $$ \int \frac{-6000} { (3x+50)^2} dx$$
2
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2answers
31 views

What's my mistake in this integral transformation?

I've the following integral, which should result in 1, as shown by the scetch, but in my calculation I get the result 0. What's my mistake? Sorry the comments are in German and please note that a ...
0
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0answers
14 views

Drawing slices and projections of an iterated integral.

I'm having a rough time visualizing and graphing the slices and projections of this iterated integral: $\int \limits _0 ^1 \int \limits _y ^1 \int \limits _y ^x x \mathbb e ^{z^2} \space \mathbb d x ...
2
votes
1answer
33 views

Understanding the Definition of a derivative as slope of a tangent line

I'm trying to understand the derivative and am wondering why the derivative is described as the slope of the tangent line and not the slope of a function itself. Say $f(x) = 2x+5$ where ...
1
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1answer
46 views

Integration by parts tricks [on hold]

Are there any useful tricks to integration by parts apart from: $$\int f(x) = \int 1 \cdot f(x)$$ $$\int f(x) = ... = g(x) + c \cdot \int f(x) \rightarrow \int f(x) = \frac{g(x)}{1-c}$$ How would ...
3
votes
3answers
72 views

Solving with integration by parts: $\int \frac 1 {x\ln^2x}dx$

Solving: $$\int \frac 1 {x\ln^2x}dx$$ with parts. $$\int \frac 1 {x\ln^2x}dx= \int \frac {(\ln x)'} {\ln^2x}dx \overset{parts} = \frac {1} {\ln x}-\int \frac {(\ln x)} {(\ln^2x)'}dx$$ $$\int ...
0
votes
1answer
30 views

Understanding Dirac Delta

I found this: here: http://www.nada.kth.se/~annak/diracdelta.pdf on page 2 Can anyone explain how and why all the terms are cancelled in the second step?
1
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1answer
15 views

$f$ is bounded by $M$ on $[a, b]$ and if the restriction of $f$ to every interval $[c, b]$ where $c$ in $(a, b)$ is Riemann integrable

If $f$ is bounded by $M$ on $[a, b]$ and if the restriction of $f$ to every interval $[c, b]$ where $c$ in $(a, b)$ is Riemann integrable, then $f$ is Riemann integrable and that $\int _c^b f \to ...
2
votes
1answer
28 views

How do I complete this proof that the absolute value of an integral function is an integrable function?

I'm trying to complete the proof in this answer that if $f: [a, b] \to \mathbb{R}$ is a Riemann integrable function, then $|f|$ is an integrable function also. I understand the proof that $$ ...
1
vote
1answer
20 views

How we can find $A_{(\Gamma_f)}$?

We have $f,g:[-4,4]\rightarrow\mathbb{R}$, $f(x)=x^2+2$ and $g(x)=x+4$. We need to find the crowd area between the graphs f and g. I know that $A_{(\Gamma_f)}=\int_a^b|f(x)|dx$ but in this case how ...
0
votes
1answer
19 views

Is the product of a Schwartz function and a locally integrable function integrable?

Let $f\in S(\mathbb{R}^n)$ the space of rapidly decreasing functions on $\mathbb{R}^n$ and $g\in L_{loc}^1(\mathbb{R}^n)$. Is $fg$ integrable? Namely is it true that $$ \int |fg| <\infty. $$