All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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60 views

Getting the bound $\frac{1}{h}\int_0^{T-h}\int_t^{t+h}\int_\Omega |\nabla u(\tau)| |\nabla u(t+h) - \nabla u(t)|\;dxd\tau dt \leq C$

Let $u \in L^2(0,T;H^1(\Omega)) \cap L^\infty(0,T;L^2(\Omega)).$ Is it possible to find the following bound: $$\frac{1}{h}\int_0^{T-h}\int_t^{t+h}\int_\Omega |\nabla u(\tau,x)| |\nabla u(t+h,x) - ...
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0answers
35 views

Indefinite Integral similar to a Gaussian

I would like to do the following integral: $$ \int_0^t \frac{1}{(2\pi (w^2+4D^2(t-t')^2))^{(3/2)}}\exp\left(-\frac{x^2+y^2+(z-vt')^2}{2(w^2+4D^2(t-t'))}\right)dt', $$ where w,D,t,x,y are all real ...
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1answer
185 views

Indefinite integral $ \int \cos x \sqrt{\cos2x}dx$

$$ \int \cos x \sqrt{\cos2x}dx $$ I know that $\cos 2x = 1-2\sin^2x,$ but not sure if it will help me? Integrate by parts? $$ \sin x \sqrt{1-2\sin^{2}x}-\int \sin x \frac{-\sin2x}{\sqrt{\cos2x}}dx ...
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0answers
104 views

Indefinite Integral

I tried to solve this indefinite integral $$\int\frac{1}{1+\tan^{-1}x}dx$$ I try taking the change of variable $u=\tan^{-1}x$ but I fail to reach a solution. Can anyone help me. Thanks in advance.
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2answers
130 views

What does Riemann-Stieltjes integral calculate when $\alpha(x) \neq x$?

When we get Riemann-Stieltjes integral becomes standard Riemann integral which calculates area under the curve. We have that $$ s(f,\alpha,P)=\sum_{k=1}^nm_k\Delta\alpha_k \ \text{ and }\ ...
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1answer
94 views

Integral of Inverse of Log X

What is the value of $$\int\dfrac{1}{\log x}dx$$ I have tried many times, but failed everytime. Can anyone help me out in solving this question.
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27 views

This shows all double integrals are zero; spot the mistake

Say we have some integral $\int_0^{\pi}d\phi_1\int_0^{\pi}d\phi_2f(\phi_1,\phi_2)\ .$ If I make the substitution $\psi_1=\phi_1-\phi_2$ and $\psi_2=\phi_1+\phi_2$, using the determinant of the ...
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1answer
276 views

Formula for $\int_0^\infty \frac{\log(1+x^2)}{\sqrt{(a^2+x^2)(b^2+x^2)}}dx$

Is it possible to express the following integral in terms of known special functions? $$I(a,b)=\int_0^\infty \frac{\log(1+x^2)}{\sqrt{(a^2+x^2)(b^2+x^2)}}dx$$ I have managed to solve the special ...
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0answers
18 views

Double integral units

I was just doing some problems and was wondering whether the answer to the following question will have any units: I was thinking it should be units^3 since it represents volume above R below xy^2. ...
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1answer
21 views

solid bounded by the plane

Suppose R is the solid bounded by the plane z = 4 x, the surface z = x^2, and the planes y = 0 and y = 3. Write an iterated integral in the form below to find the volume of the solid R. I need ...
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2answers
30 views

integral to find the volume of the solid

Use a triple integral to find the volume of the solid bounded by the parabolic cylinder and the planes and
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1answer
44 views

triple integral [closed]

Evaluate the triple integral $$\int\int\int_V xy dV$$ where E is the solid tetrahedon with vertices $$(0,0,0), (8,0,0), (0,8,0), (0,0,7)$$
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1answer
31 views

mass of the solid

Find the mass of the solid bounded by the $xy$-plane, $yz$-plane, $xz$-plane, and the plane $x/2+y/3+z/6=1$ if the density of the solid is given by $\delta(x,y,z)=x+4y$. Cannot set the limits and the ...
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2answers
53 views

triple integral bounded by cylinder

Evaluate the triple integral where E is the solid bounded by the cylinder and the planes and in the first octant. I tried really hard to get the write answer but I am still struggling in the ...
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1answer
70 views

Volume of a solid

Find the volume of the solid in $\Bbb R^3$, bounded by $$y = x^2\\x=y^2\\z=x+y+30\\z=0 $$ For me setting the integral is the issue!
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1answer
70 views

Antiderivatives…relations?

Is their any correlation between a function and it's antiderivative? I know the formula for to get said antiderivative is $\int F(x)dx=f(x)+C$ Where F(x) is the antiderivative of f(x) and where C ...
20
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2answers
479 views

Integral $\int_0^1\frac{\ln x}{x-1}\ln\left(1+\frac1{\ln^2x}\right)dx$

Is it possible to evaluate this integral in a closed form? $$ I \equiv \int_{0}^{1}{\ln\left(x\right) \over x - 1}\, \ln\left(1 + {1 \over \ln^{2}\left(x\right)}\right)\,{\rm d}x $$ Numerically, ...
3
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1answer
142 views

prove $\int_{0}^{2\pi}(1-\cos x)^n\cos nx dx=(-1)^n\frac{\pi}{2^{n-1}} $

prove $$\int_{0}^{2\pi}(1-\cos x)^n\cos nx dx=(-1)^n\frac{\pi}{2^{n-1}} $$ I tried with $2\cos x =z+\frac1z$ then use residue theorem but I faced some troubles my try is : $2\cos x =z+\frac1z$ ...
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0answers
47 views

Integration Question - Not sure how to approach

I have absolutely no idea how to approach this question: $$\int \frac{x^2}{(15+6x-9x^2)^{3/2}} \ \mathrm{d}x$$ I'm almost positive that it has something to do with trigonometric substitution, but ...
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1answer
35 views

Partial differential equations help

Can some one please, help me to solve this exercise : Let $A$ be an open of $R^{d}$, (d>1), and $f$ a real continuous valued over $A$. Show that if for all $ v\in\mathcal{C}^{1}(A)$, $\int_{A}fv ...
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1answer
123 views

Analytical Formula for Hilbert Transform of a Ricker Wavelet

I am attempting to validate some numerical code I have to compute Hilbert transforms. As I am interested in the Hilbert transforms of functions with rapid decay, I wanted to unit test my code with the ...
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2answers
109 views

Integration with substitution $u=\cos(x/2)$

Please could you help me solve this integral Find $$ \int \cos x \, \sqrt{1-\cos x} \, dx.$$ Hint: use the substitution $u=\cos(x/2)$. Thanks.
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1answer
32 views

Integrating the difference of brownian motion

I'm reading the solutions to an exercise where it is stated that $$\int_t^T\Big(W(u) - W(t)\Big)du = \int_t^T (T-u)dW(u).$$ But can someone enlighten me to what theorem/rule can be used to show this? ...
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2answers
59 views

Evaluating an integral in calculus III

Please can someone explain what happened after step 3 , in this upload image of the exercise Please make it clear how the "1/2" came in step 4 , and why we are subtracting the integrals
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0answers
10 views

Correlation 4-point

I need to calculate $\langle x_{i}x_{j}x_{k}x_{l}\rangle $, where $$ \langle f(x) \rangle = \int e^{-\frac{1}{2}A_{ij}x^{i}x^{j} - \frac{\lambda }{4!}\sum_{i}x_{i}^{4}} f(x)d^{n}\mathbf x , $$ for ...
1
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1answer
139 views

Compute the volume bounded by a parametric surface

Given a smooth (i.e. $C^1$ continuous) closed parametric surface $(x,y,z) = S(u,v)$, how do I compute the volume $\mathcal{V}$ bounded by it? After browsing through a couple of books, I'm sure it is ...
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2answers
157 views

Value of this definite integral $\int_{0}^{\infty} \frac{ \ln(x)}{x^2+2x+4} dx $

So I came across this question on brilliant.org and didn't know how to go about it: $$\int_{0}^{\infty} \frac{ \ln(x)}{x^2+2x+4} dx $$ I tried to complete the squares in the denominator and then use ...
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1answer
77 views

$\int_0^{1/2} \cos^{-1} x \, dx$ using integration by parts.

integral lower bound is 0. upper bound is 1/2. function is cos^-1 x dx. My work: I use integration by parts. u = cos^-1 x ... du = -dx/sqrt(1-x^2) ... v = x ... dv = dx ... so integral udv = ...
8
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1answer
178 views

Closed Form for $\int_0^1 \frac{\log(x)}{\sqrt{1-x^2}\sqrt{x^2+2+2\sqrt{2}}}dx$

Is there a closed form for the following integral? $$\int_0^1 \frac{\log(x)}{\sqrt{1-x^2}\sqrt{x^2+2+2\sqrt{2}}}dx$$ It is approximately equal to $-0.48878092308456029189008$. Mathematica is ...
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0answers
34 views

Solving a system of nonlinear equations involving summation, integration and squares

I can only get implicit results for these equations. I looked everywhere but I wanted to check whether you can see something I can't. I find implicit results for all of the variables. I need to find ...
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0answers
64 views

On $\int_{-\infty}^{+\infty} {\frac{\tan(t-t_0)}{\cosh^2(t-t_0)} \cos(\omega t) \,\mathrm{d}t}$

How to count this? $$ \int_{-\infty}^{+\infty} {\frac{\tan(t-t_0)}{\cosh^2(t-t_0)} \cos(\omega t) \,\mathrm{d}t} $$ Can we use residue formula?
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16 views

Representation theorem on normal spaces

I only know the representation theorem on locally compact Hausdorff spaces. I heard that there is a normal space version in the book of Dunford & Schwartz. However I cannot find where it is. Can ...
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1answer
109 views

Integral $ \int_{-\infty}^\infty \frac{e^{ikx}}{x^{3/2}}dx$

Hi I'm trying to solve this integral Fourier Transform $$ \int_{-\infty}^\infty \frac{e^{ikx}}{x^{3/2}}dx=\sqrt{2\pi|k|}(1+i) (-1+\text{sgn}(k)) $$ where sgn(k)$=1$ for k>1 and $-1$ for k<1. I am ...
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1answer
21 views

Curious integral pattern

While messing around with integrals and testing different values, I observed that the following seems to hold for any two functions $f(x),g(x)$: $\int_0^L f(x)g(x)\,dx=\int_0^{L/2}2f(2x)g(2x)\,dx$ ...
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0answers
51 views

Integral of P(x)/(A(x)E(x)

I have a question where I have to use the formula $\delta = \int_0^L \frac{P(x)}{A(x)\epsilon(x)}$. It is used to find the 'displacement' or the 'elastic deformation' of an axially loaded member like ...
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1answer
24 views

Is there any integration of a defined function that could not be express as a convergent infinite series?

Is there any integration of a defined function that could not be express as a convergent infinite series?Like if it could be getting a divergent series as an answer I wonder if the answer is a yes or ...
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1answer
39 views

Lifetime of exponential variable of a battery

Suppose that the operating lifetime of a certain type of battery is an exponential random variable with parameter $\theta=2$ $($measured in years$)$. Find the probability that a battery of this type ...
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0answers
31 views

why we say this function have closed form while the other doesn't?

why we say this function have closed form while the other doesn't? $\int(sin(x)) dx=-cos(x)+ C$ have a closed form While $\int sin(x)/x$ dx = Si(x)+ C does not have a closed form?
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54 views

What is list of common integral that have no closed form?

What is list of common integral that have no closed form? It's diffucult for me to google it for some reason.
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1answer
39 views

Do I have to use this formula to calculate the integral?

I have the following exericse: "Calculate the integral $$\oint_C{(x+y)}ds$$ where $C$ is the line segment $x=t, y=1-t, z=0$, from $(0,1,0)$ to $(1,0,0)$." $$$$ To calculate this integral do I have to ...
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1answer
39 views

Are the $L^p$ norms ordered by $p$?

A question left over from this post is: Are the $L^p$ norms ordered by $p$ like the power means are?
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31 views

Lifetime of pdf disk

The pdf for the lifetime X, in years, of a Superstuff disk drive is given as follows: $f(x) = \begin{cases} 2/x^2 & \text{for } x\geq2\text{ } \\ 0 & \text{elsewhere} \end{cases}$. ...
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0answers
39 views

Performance estimation of shellSort

I'm trying to make a performance estimation for shell-sort algorithm. And I fail in it. My formula: equals to where dz is outer while-loop, dy is middle for-loop, and dx is inner for-loop ...
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0answers
22 views

Copying the Curvature of One Function onto Another: Approximation

I have a polar function $$ r(\theta)=\left(r+\epsilon\right)\cos(\theta)-\sqrt{r^{2}-\left(r+\epsilon\right)^{2}\sin^{2}(\theta)} $$ Is it possible to methodically conjure another polar function ...
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1answer
673 views

Integral $\int_{-\infty}^\infty J^3_0(x) e^{i\omega x}\mathrm dx $

Hi I am trying to evaluate the integral $$ \mathcal{I}(\omega)=\int_{-\infty}^\infty J^3_0(x) e^{i\omega x}\mathrm dx $$ analytically. We can also write $$ ...
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2answers
59 views

Evaluating the integral ${\int_0^{1000} e^{x-[x]}dx}$

Evaluate the integral: $${\int_0^{1000} e^{x-[x]}dx}$$ This is what I have tried to do: $$\int_0^{1000} e^{x-[x]} dx = 1000 \times \int_0^1 e^{x-[x]} dx.$$ Next, $1000 \times {1.(1-1)} = ...
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0answers
47 views

integral involving hypergeometric function

I've obtained that the eigenfunctions of a certain Sturm-Liouville problem are: $$ \phi(x,\lambda) = C\cdot(1/x)^{-1/2\pm i\lambda}\Psi(-1/2\pm i\lambda, 1\pm2i\lambda,1/x), $$ where $C$ is a ...
1
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1answer
58 views

Differentiability of function defined as integral

Suppose $$F(x) := \int_0^1 f(t,x) dt $$ is well-defined for all $x \in \mathbb{R}$. I would like to show that $F(x)$ is not differentiable at $0$. Is it enough to show that $\partial_x f(t,0)$ is ...
1
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1answer
31 views

Solving integral including a triangle

How can I solve this integral? Image link: http://oi61.tinypic.com/2jeoga1.jpg I tried to solve it: x^2/2 from 4 to 0. [(4^2/2)-(0^2/2)]=8 but its wrong. Do I have to multiply base*height/2 because ...
2
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0answers
30 views

Using Polars to Approximate a Cartesian line: Approximating an Integral

I have the equation of the lower semicircle of radius $r$ centred at a distance $a+r$ above the x-axis $$ f(x)=r+a-\sqrt{r^{2}-x^{2}} $$ which I can approximate (for small $x$) as $$ f(x)\approx ...