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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0
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0answers
7 views

Abel inversion where axisymmetric function is multiplied by $cos(\phi)$

I have a problem seems similar to Abel inversion, but the axisymmetric function is multiplied by $\cos{\phi}$, making the integrand non-axisymmetric. Here is a picture of the problem: Each chord is ...
0
votes
0answers
8 views

Restriction over pdf such that an integral inequality holds $\int_{-\infty}^{+\infty}\left(F(x)-\frac{2}{3}\right)xf(x)dx\geq 0$

Let $f(x)$ be a pdf in $(-\infty,+\infty)$ and $F(x)$ it's cdf. Assume both are smooth. I need to find restrictions over the pdf such that the following inequality holds: ...
0
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0answers
28 views

Solving an integral that includes an exponential function and the error function

This question contains all the values needed to compute an equation. My question is, do you get the same result I get? Or do you get the result in the paper I've linked to? I'm trying to decipher ...
-2
votes
2answers
59 views

Taylor Series for $\frac{1}{ 1+x+x^2}$

I tried to solve it in a way. The solution did not match. Please tell me where i went wrong. $\cfrac {1} {1+x+x^2} = \cfrac 4 {4+4x+ 4x^2} = \cfrac 4{ 3+(2x+1)^2} = \cfrac 1{\sqrt 3}\cdot\cfrac 4{ 1+ ...
0
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0answers
19 views

How to obtain accumulated counts of past events by time $t$?

Given $f: [0, \infty) \to \{0,1\}$, $f(t)$ represents whether there is an event occurring at time $t$. How can we obtain $g: [0,\infty) \to \mathbb{N}_0$ so that $g(t)$ represents the number of ...
1
vote
3answers
25 views

Integration of the square root of a quadratic

I am in the tricky situation of trying to integrate the following. $$\sqrt{4 a^2 (y-b)^2+c^4}$$ $a, b$ and $c$ are all known constants. Can anybody provide insight as to how to do this? I have ...
2
votes
2answers
68 views

Result of $\int \limits_{-\infty}^{+\infty}x^2\times\exp\left(\dfrac{-x^2}{2}\right)\mathrm{d}x$ [duplicate]

I would like to read a very thorough and explained calculation process for a couple of integrals. For the life of me I just can't figure out the result on my own, and no resource on the web were able ...
4
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0answers
62 views

Another integral related to Fresnel integrals

How would we prove this result by real methods ? $$\int_0^{\infty } \frac{\sin \left(\pi x^2\right)}{x+2} \, dx=\frac{1}{4} \left(\pi-2 \pi C\left(2 \sqrt{2}\right)-2 \pi S\left(2 ...
1
vote
4answers
39 views

Integrating linear/trigonometric

I have the following question- $\int$ $\frac{x}{1+cosx}dx$ Do I do integration by parts or is there some other method? Thanks for the help.
2
votes
2answers
26 views

Integration of a scalar function with respect to a vector

I have a scalar function that takes $n$ arguments, $f(x_1, x_2,x_n) = f(\mathbf{x})$, and two vectors also with $n$ elements, $\mathbf{z} = (z_1, z_2\cdots,, z_n)$, and $\Delta\mathbf{z} = (\Delta ...
-4
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0answers
56 views

Hardest integration ever? [on hold]

integrate $\int \frac{1}{1+x^4 }dx$ I have no idea how to do this, I have a test tomorrow pls help! I tried adding and subtracting $2x^2$ and use $a^2-b^2$.
1
vote
1answer
94 views

$\int_0^1(1+\log(x))\sin(x)dx$ How to solve this Integral?

$$\int\limits_0^1(1+\log(x))\sin(x)dx$$ Someone has challenged me to solve this, I solved it without bounds, I have no idea how to do it with those limits.. Is $u=1+\log(x)$ right substituion? or ...
0
votes
0answers
28 views

Differentiating CDF

I'm trying to differentiate the cdf of z with respect to x where the upper bound is a function of x and z ~ N(a , $b^2$ $\cdot$ $x^{-2}$) $\frac{d}{dx} \int _{-\infty} ^ {z^*(x)} \Phi ^{\prime} (z) ...
2
votes
2answers
99 views

How to solve $ \int \limits _0 ^{2\pi} \dfrac{dx}{(\alpha +\beta\cos x)^2} $

I am trying to solve this integral, I think that it could be solve using the complex. $$ \int \limits _0 ^{2\pi} \dfrac{dx}{(\alpha +\beta\cos x)^2} $$
0
votes
1answer
40 views

Differentiation Commute with Lebesgue Integration

My question is simple: Given $f: \mathbb{R}^{n+m} \to \mathbb{R}$, $f\in C^{k}(\mathbb{R}^{n+m})$ , and $X \subset \mathbb{R}^{n}$. Write $f$ as $f(x_1, \ldots, x_n, t_1, \ldots t_m)$. When is ...
0
votes
3answers
48 views

Intuitive meaning of the probability density function at a point

I understand how to integrate probability density functions to find probability within a certain range. However, what I don't understand is what it would mean to set the variable (say x or y) to a ...
0
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0answers
34 views

Is it possible for the Simpson's method to converge faster than Rombergs method?

I have the following integral: $\int_{0}^{100} \frac{x^{3/2}}{\cosh{(x)}}dx$ I am running code for the Simpson's method and Romberg method to evaluate the integral numerically and the results show ...
2
votes
3answers
93 views

Primitive of $\int { \frac { x^{ 2 } }{ (x\sin x+\cos x)^{ 2 } } dx } $

How do I evaluate the integral of $$\int { \frac { x^{ 2 } }{ (x\sin x+\cos x)^{ 2 } } dx } $$ in a simple way? The way I could do the question, was by multiplying and dividing the fraction by $\cos ...
3
votes
3answers
55 views

Reduction formulae in definite integration

$$I_n = \int_0^{\pi}\frac{\sin^2(nx)}{\sin^2(x)}dx $$ Find relation between $I_n$, $I_{n+1}$ and $I_{n+2}$ I tried integration by parts by taking $\sin^2(nx)$ as the first function, but reached ...
3
votes
1answer
93 views

Calculating $\int_0^{\pi/4} \frac{\cot (x)}{\cot ^2(x)+\sqrt{\cot (x)}} \, dx$

This is not really one of that kind of integrals that Mathematica cannot handle with, but given the case of a contest, how would we like to handle with it? I would like so much to know your ideas ...
0
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0answers
26 views

Integral of exponential complex trigonometric functions

I have a problem with this integral: $X_{e11}$ = $\int_{0}^{2\pi} \int_{0}^{\pi}e^{-ikr_{n}\left(\sin\vartheta \cos \varphi \sin \theta_n \cos \phi_n + \sin \vartheta \cos \varphi \sin \theta_n \sin ...
0
votes
2answers
30 views

Why is the estimate of the order of error in Trapezoid converging to $2.5$?

The integral in question is: $\int_{0}^{\infty} \frac{x^{3/2}}{\cosh{(x)}}dx$ I coded a program to compute $p$, an estimate of the order of the error for the Trapezoid method of numerical ...
0
votes
1answer
20 views

Evaluating This Complex Line integral

I'm trying to evaluate the following: $$\int_{\mathcal{C}}z^3 e^{-z^4}\,dz $$ along the path $\mathcal{C}=\left\{\sin(t^2)-i\frac{2t^2}{\pi}:0\leq t\leq\sqrt{\frac{\pi}{2}}\right\}.$ I tried using ...
1
vote
1answer
49 views

Definite Integration [on hold]

For some constant c, we wish to compute the following integration (or a tighter bound on the same) $\int_{\theta}^{1} x \exp \left(- \frac{c\theta}{x}\right) dx $
1
vote
1answer
64 views

Closed form of this sum

$$\sum _{ s=1 }^{ \infty }{ \left( \frac { 1 }{ 4s-1 } \sum _{ n=0 }^{ \infty }{ \left( \frac { 1 }{ n+1 } \sum _{ k=0 }^{ n }{ \left( \left( \begin{matrix} n \\ k \end{matrix} \right) \frac { { ...
1
vote
0answers
35 views

Integrals with error function and exponentials

I'm trying to solve the integrals below: $$\int_{-\infty}^\infty \int_{-\infty}^\infty \frac{x}{\sqrt{x^2+y^2}}\cdot \operatorname{erf}\left(m\cdot\sqrt{x^2+y^2}\right) \cdot \exp(-a\cdot ...
1
vote
3answers
50 views

Limits using definite integration

$F(k)$ = $$ \lim_{n\to \infty}{\frac{1^k + 2^k +...+n^k}{(1^2 + 2^2 +...+n^2)*(1^3 + 2^3 +...+n^3)}} $$ I need help in finding $F(5)$ and $F(6)$. I tried converting it into summation form and using ...
4
votes
2answers
76 views

Evaluating definite integral of $e^{i t^2}$

In passing Sakurai's QM book mentions that $$\int_{-\infty}^\infty e^{i t^2} dt = \sqrt{i \pi}$$ This is consistent with 7.4.4 in Abramowitz and Stegun which claims for $\Re a > 0, n = 0, 1, 2, ...
-3
votes
0answers
32 views

Conditions for Riemann integrability [on hold]

A function $f$ is Riemann-Integrable iff the infimum of the upper sum and the supremum of the lower sum of all partitions P of a closed interval [a,b] in the domain of $f$ coincide, as stated below: ...
0
votes
0answers
29 views

Solving for the limit of a Gaussian random variable within an integral

I'm having trouble solving a particular integral. It is $$ (1/\Delta t)\int_t^{t+\Delta t}I(t')dt', $$ where $$ I(t') = \mu_c+\sigma_c \eta(t'). $$ In this second equation, $$ \eta(t') = ...
1
vote
1answer
26 views

Integration divided by the function

How do I guarantee that $ \frac{\int_0^v f(x) dx}{f(v)} $ is increasing? Under which assumptions is this true? Or, what types of properties would such a function have? Thanks.
4
votes
0answers
77 views

Evaluating $\int_0^{\pi /2}\left(\frac{1}{\sqrt{\tan(x)}}+\frac{1}{\sqrt{\arctan(x)}}\right) dx$ [on hold]

I've come across the following integral: $$\int_0^{\pi /2}\left(\frac{1}{\sqrt{\tan(x)}}+\frac{1}{\sqrt{\arctan(x)}}\right) dx$$ I haven't been able to make any of the obvious methods work (or make ...
0
votes
0answers
24 views

integral of complex function, power series

let $\mu$ be a finite borel measure on $[0,+\infty)$ and let $f$ be defined by $$f(z)=\int_{[0,+\infty)}\frac{d\mu(t)}{t-z},\quad z \in \mathbb{C} \setminus [0,+\infty)\,.$$ *show that the integral ...
5
votes
3answers
74 views

Evaluate $\int \theta\sec\theta \tan\theta \ d\theta$

integral of $\int \theta\sec\theta \tan\theta \ d\theta$ my work $\frac{d}{d\theta}\sec(θ) = \sec(\theta)\tan(\theta)$ So if we let $u = \theta$ and $v' = \sec(\theta)\tan(\theta)$, then we get: ...
3
votes
2answers
91 views

solve for $\int_{0}^{{\alpha}{b}}(a^x-1)dx=\int_{{\alpha}{b}}^{b}(a^x-1)dx$

I am sitting with a problem and my calculus is a bit (ok very) rusty. $\int_{0}^{{\alpha}{b}}(a^x-1)dx=\int_{{\alpha}{b}}^{b}(a^x-1)dx\\ 0<\alpha<1\\ b\geq1$ Solve for a. any help would be ...
3
votes
1answer
37 views

bounding a sum using a definite integral

Conjecture. Let $1<p<\infty$. Then there exists $C\in(0,\infty)$ such that for any $k\in\mathbb{Z}^+$ we have \begin{equation}\sum_{n=1}^k(k+1-n)^{-\frac{p}{p+1}}n^{-\frac{1}{p+1}}\leq ...
0
votes
0answers
7 views

Deriving the integration limits after a non-standard change of variables in a triple integral

Is there an algorithm for deriving the integration limits after a non-standard change of variables in a triple integral? I have a rather complicated triple integral to perform. Some details: the ...
1
vote
2answers
30 views

Evaluating an improper integral containin exp and sqrt

Is it true that $\int_0^\infty y^{-1/2}e^{-\lambda y}dy=\sqrt{\frac{\pi}{\lambda}}$? I'm not sure if the integral exists for all $\lambda$ or is the value correct. It looks hard for me to evaluate ...
3
votes
1answer
53 views

Find: $\lim_{n \to \infty} \int_0^{\infty} \arctan(nx) e^{- x^n}dx$

Find: $$\lim_{n \to \infty} \int_0^{\infty} \arctan(nx) e^{- x^n}dx$$ Probably, no recursive form could be found, and elementary tools (integration by parts, change of variable, etc.) are ...
-3
votes
0answers
22 views

Trapz vs gausslegendre - integration methods [on hold]

I have a code that uses trapz. But i want to try gausslegendre, but its different method and its not easy to change. Any help ? ...
2
votes
2answers
44 views

bend measurement and calculating $\int_4^8 \sqrt{1+{\left(\frac{{x^2-4}}{4x}\right)^2}} $

How can i get the measure of this bend : $y=\left(\frac{x^2}{8}\right)-\ln(x)$ between $4\le x \le 8$. i solved that a bit according to the formula $\int_a^b \sqrt{1+{{f'}^2}} $:$$\int_4^8 ...
6
votes
1answer
110 views

Evaluating $\int\sqrt{\frac{1-x^2}{1+x^2}}\mathrm dx$

Evaluating $$\int\sqrt{\frac{1-x^2}{1+x^2}}\mathrm dx$$ I had read the similar problem, but it doesn't work.
0
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1answer
37 views

Complete the square doesn't work here.

In this substitution, I tried the method of complete the square, but at the end the answer is not correct. I'm totally sure that the answer is: ...
0
votes
0answers
16 views

Need Help With this Integral:

I am working on a probability problem and I have figured everything out except I am having trouble calculating this variance: $\int_{-\infty}^{\infty} (t-s)_+^d-(-s)_+^d ds$ and I was wondering if ...
1
vote
1answer
53 views

How is this trival?

I was reading an article today and on section 2 it is indicated that if we are given a Radon Measure $\mu$, and a real $p$ then fast convergence entails trivially almost sure convergence, where fast ...
0
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0answers
17 views

How to choose grid for a numerical integral of complex function?

I need to numerically integrate a complex function $f(x)$ on R, i.e. to approximate $\int_{-\infty}^\infty{f(\xi)d\xi}$. Performance is crucial as the integration is repeated a high number of times ...
0
votes
0answers
18 views

Does this theorem concerning upper and lower bnound of a monotone decreasing function have a formal name?

This is the theorem: Let $g$ be a monotone decreasing function and let $a,b \in \mathbb{N}$. Then the following holds true: $$\int_{a}^{b+1}g(x)dx \overset{(i)}{\leq} ...
10
votes
3answers
158 views

Calculating $\int_0^{\infty } \left(\text{Li}_2\left(-\frac{1}{x^2}\right)\right)^2 \, dx$

Do you see any fast way of calculating this one? $$\int_0^{\infty } \left(\text{Li}_2\left(-\frac{1}{x^2}\right)\right)^2 \, dx$$ Numerically, it's about $$\approx ...
6
votes
3answers
118 views

Closed-form of $\operatorname{Li}_2\left(1 \pm i\sqrt{3}\right)$

I've found the following identity while I was going through a quite difficult path. $$ \Re\operatorname{Li}_2\left(1 \pm i\sqrt{3}\right) = \frac{\pi^2}{24} -\frac{1}{2}\ln^2 2 - ...
0
votes
0answers
12 views

General case for differentiation under the integral sign

What is the most convenient way to decide if we can differentiate under the integral sign? If the integrant is a smooth function, could we do so?