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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2
votes
1answer
48 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
0answers
31 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
44 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
votes
0answers
27 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
82 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
46 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 ...
2
votes
1answer
68 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
votes
0answers
25 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
19 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
46 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
54 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
31 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
49 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
27 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
74 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
19 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
73 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
88 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
36 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
43 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
108 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
votes
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
52 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
votes
0answers
16 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
150 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
111 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?
12
votes
3answers
131 views

Prove that $\int_0^1 \frac{1}{1+\ln^2 x}\,dx = \int_1^\infty \frac{\sin(x-1)}{x}\,dx $

I've found the following identity. $$\int_0^1 \frac{1}{1+\ln^2 x}\,dx = \int_1^\infty \frac{\sin(x-1)}{x}\,dx $$ I could verify it by using CAS, and calculate the integrals in term of ...
3
votes
1answer
34 views

computing an integral (fraction of real powers)

I want compute the following integral depending on the real parameters $\alpha, \beta > 0$ and $C >0$ $$ \int_0^1 \frac{u^{2\beta}}{C+u^{2(\alpha+\beta)}} du$$ Thanks a lot for any clue !
1
vote
1answer
19 views

Find the values of the derivatives of the integral with a variable inside its limits.

$\require{cancel}$ Problem: I have the function $g: \mathbb{R} \to \mathbb{R}$ defined as $$ g(x)=\int^{(1+x^2)}_{-(1+x^2)} sin(t^3)\ dt,\ x \in \mathbb{R} $$ I would like to calculate values of ...
0
votes
4answers
38 views

why $\int\frac{dx}{2\sqrt x +2x} = \ln(1+\sqrt x)$+C [on hold]

I don't know why $\int\frac{dx}{2\sqrt x +2x} = \ln(1+\sqrt x)$+C .The maple solve that like this : $$\int\frac{dx}{2\sqrt x +2x}:$$ $$=\int\frac{du}{u+1}$$ $$=\int\frac{du_1}{u_1}$$ $$=ln(u_1)$$ ...
-1
votes
0answers
24 views

How do we know which quantity to be integrated with respect to which? [on hold]

Suppose I write z = x1y1 + x2y2 + ... + xnyn So, if x and y are not discrete values, but continuous functions, then would z be written as the integral of x.dy or would it be y.dx or something else ...
2
votes
1answer
45 views

Convergence of an integration $t=\int_{x_0}^{x_1}\sqrt{\frac{m}{2(E_0-V(y))}}dy$

When I am reading Brian Hall's "Quantum Theory for Mathematicians", I came across an integration (frequently appeared in physics textbooks) $$t=\int_{x_0}^{x_1}\sqrt{\frac{m}{2(E_0-V(y))}}dy.$$ The ...
3
votes
4answers
56 views

Integration using trig substitution or substitution

I was trying to review calculus integration techniques before my differential equations class. I came across $\int \frac{1}{\sqrt{1-2x^2}}\,\mathrm{d}x$. I can't exactly figure out a good way to solve ...
2
votes
5answers
102 views

definite integral of $x^2e^{-x^2}$

I am trying to calculate the integral of this form: $\int_{-\infty}^{+\infty}e^{-x^2}\cdot x^2dx$ I am stuck. I know the result, but I'd like to know the solution step-by-step, because, as some ...
3
votes
1answer
35 views

Surface of $(x^2 + y^2 + z^2)^2 = a^2 * (x^2 - y^2)$ using surface integrals

I have to find the surface of $$(x^2 + y^2 + z^2)^2 = a^2(x^2 - y^2)$$ using a surface integral and really have no idea what to do... I would really appreciate it if you could give me an idea.
-3
votes
0answers
71 views

$\int \left(\cos\left(e^{\sin(x)}\right)\right) \text{d}x$ [on hold]

I have to find this integral but I have no idea if there is any solution to it: $$\int \left(\cos\left(e^{\sin(x)}\right)\right) \text{d}x$$
0
votes
1answer
62 views

How was this differentiated?

How red-circled function with 1/D is equal to green-circled? Note: D is equal to dy/dx. Update: Complete pic
0
votes
1answer
32 views

Line integral and checking its path independence in three dimensions

I have a following exercise, falling under the topic of line integrals. Calculate the integral: $$I=\int\limits_{\gamma} \sin(yz)\,dx+xz\cos(yz)\,dy+xy\cos(yz)\,dz$$ Where: ...
3
votes
4answers
97 views

Why $\int_{0}^{\pi/2}\tan(x/2) dx= \ln 2$ [on hold]

I don't know why $$\int_0^{\pi/2}\tan\frac{x}2\ dx= \ln 2.$$ How can i solve this to get that answer?