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 ...

learn more… | top users | synonyms (3)

0
votes
0answers
23 views

Fubini's theorem (interchange of sum and integrals) in case of multivariable function

Can the Fubini's theorem in case of single variable sequence of functions be readily extended to multivariable sequence of functions?, i.e, Is it true to say $$\iiint_V\sum_{n=0}^\infty f_n(u,v,w) \,...
1
vote
0answers
18 views

Kline Calculus intuitive approach Chapter 3 problem 12

The problem is as follows : Water drops flow out from a small opening at the rate of one drop per second and fall vertically with an acceleration of 32 ft/sec^2. Determine the distance between two ...
1
vote
1answer
24 views

Evaluate the integral $\int_{-\infty}^{\infty}{\left| 2t\cdot\text{sinc}^2(2t)\right|^2}\,dt$

I have a question in solving the integral $$\int_{-\infty}^{\infty}{\left| 2t\cdot\text{sinc}^2(2t)\right|^2}\,dt.$$ I know that you can use Parseval's Theorem to prove that $\int_{-\infty}^{\infty}\...
4
votes
3answers
168 views

Integration by parts or substitution?

$$\int_{}^{}x e^x \mathrm dx$$ One of my friends said substitution , but I can't seem to get it to work. Otherwise I also tried integration by parts but I'm not getting the same answer as wolfram. ...
0
votes
1answer
46 views

Compute definite integral by hand [on hold]

How can I compute $$\int_0^1 \frac{x^3t}{(x^2+t^2)^2} \, \mathrm{dt}$$ by hand?
1
vote
2answers
30 views

Improper integral - checking convergence of $\int_{1}^{\infty} x^2 \sin(x^4) dx$

Does the following improper integral converges ? $$\int_{1}^{\infty} x^2 \sin(x^4) dx$$ Tried to find some known improper integral to compare this one to, but didn't find one. Thanks for helping!
2
votes
1answer
75 views

How to prove that$\int_{0}^{1}\ln{(x/(1-x))}\ln{(1+x-x^2)}\frac{dx}{x}=-\frac{2}{5}\zeta{(3)}$

$$\int_{0}^{1}\ln{(x/(1-x))}\ln{(1+x-x^2)}\frac{dx}{x}=-\frac{2}{5}\zeta{(3)}$$ Put $$\frac{x}{1-x}=y$$ $$I=\int_{0}^{\infty}\ln{y}\ln{(1+3y+y^2)}\frac{dy}{y(y+1)}=\frac{8}{5}\zeta{(3)}$$ Simple ...
1
vote
2answers
51 views

Given $f(x,y)$ is a continuous function, Do these integrals equal? [on hold]

Given range $\{ 0 \le x \le 1, 0 \le y \le 1\}$ Do these integrals equal? $\int_0^1(\int_0^y f(x,y)dx)dy = \int_0^1(\int_0^x f(x,y)dy)dx$ Well, the answer is no. It seems like the triangulars are ...
1
vote
2answers
54 views

Closed form for an integral with log and power

Let $n \in \mathbb{N}$. We know that: $$\int_0^1 x^n \log(1-x) \, {\rm d}x = - \frac{\mathcal{H}_{n+1}}{n+1}$$ Now, let $m , n \in \mathbb{N}$. What can we say about the integral $$\int_0^1 x^n \...
0
votes
0answers
10 views

numerical integration asymptotic relation

Let $Q\subset R^n$ be a convex subset and $f\in C^2(Q)\;$ We set $x_s:=\int_Q xdx$,$\;\;\;Vol(Q):=\int_Q 1dx$ and $diam(Q)=sup||x-y||_2$ Prove the following asymptotic relationship: $...
5
votes
1answer
96 views

Evaluate $\int \frac {\sin(x)}{x^2 + 4x + 5}dx$

Question: Evaluate $$ \int \frac{\sin(x)}{x^2 + 4x + 5} dx=\int \frac {\sin(x)}{(x + 2)^2 + 1}dx $$ By using the change of variable $y = x + 2$ we have that $dy = dx$ then $$I = \int \frac{\...
2
votes
3answers
50 views

Showing that $\int_{-n}^{n}{x+\tan{x}\over A +B(x+\tan{x})^{2n}}dx=0$

Where n is an integer, $n\ge1$ and $(A,B)$ just constants $$I=\int_{-n}^{n}{x+\tan{x}\over A +B(x+\tan{x})^{2n}}dx=0$$ It is obvious that $$\int_{-n}^{n}x+\tan{x}dx=0$$ Let make a ...
-4
votes
0answers
27 views

Tuition service for math CBSE [on hold]

Can anyone suggest best tuition service for CBSE 12th grade math in Kuwait?
-1
votes
5answers
78 views

Evaluation of $\int_{-1}^{0}\frac{x^2+2x}{\ln(x+1)}dx$

Evaluation of $\displaystyle \int_{-1}^{0}\frac{x^2+2x}{\ln(x+1)}dx$ $\bf{My\; Try::}$ Let $$I = \int_{-1}^{0}\frac{x^2+2x}{\ln(x+1)}dx\;,$$ Put $x+1=t\; $ Then $dx = dt$ and changing limits, we get ...
1
vote
0answers
19 views

Matlab double integration result does not match with my self calculate

Here is a double integration, self learning $$ P=\int_{-w}^{w}\int_{l}^{\frac{y_h(x_b+w)}{x_h}+l}\frac{1}{2}\operatorname{erfc}\left[\frac{\log{\frac{z_h(y_b-l)}{y_h}}-\mu}{\sigma\sqrt2}\right]\space ...
-2
votes
1answer
37 views

What are the solutions to integration problems below? [on hold]

$$\int (a^2 - y^2)y dy $$ $$\int \frac{e^{\sqrt{x}} + 1}{\sqrt{x}} \ dx$$ $$\int \frac{x^3}{\sqrt{1-2x^2}} \ dx $$
0
votes
0answers
33 views

show the if f(x)'>0 the integral exist, help please. [on hold]

I need to show that if f(x) is Superior monochrome in [a,b] the integral for f(x) in [a,b] exist, from here I know that if f(x) is Superior monochrome for all x2,x1∈[a,b], (x2>x1), f(x2) ≥ f(x1) I ...
1
vote
0answers
29 views

Parallelizable open dense subset and integration

In Petersen's Riemannian Geometry (2016), it is stated on page 8 that any manifold $M^n$ has an open dense subset $O$ with $TO=O\times\Bbb R^n$. Thus it is orientable and one may define the integral ...
0
votes
3answers
54 views

How to find a parametrization for a torus?

I need to compute the surface area of the torus $$T^2=\{(x,y,z)\subseteq\mathbb R^3 \left(\sqrt {x^2+y^2}- R\right)^2+z^2=r^2\}$$ where $0<r<R$. I know I need to compute the metric tensor and ...
2
votes
1answer
26 views

Integrating a composition

I'm need to calculate this: $$\int g'(x) (2g(x) - \frac{1}{g^2(x)}) dx $$ I think i have to integrate by parts, so i put: $$ dv= g'(x) dx,v=g(x)$$ $$u=2g(x) - g(x)^{-2},du=2g'(x)-(g(x)^{-2})'$$ ...
-1
votes
0answers
41 views

Prove for integrals the following [duplicate]

If n is a positive integer, prove that $\int_0^n{[t]^2 dt} =\frac{n(n-1)(2n-1)}{6}$. Please prove it in a descriptive way.
1
vote
1answer
42 views

Produce a sequence $(g_n):g_n(x)\ge 0$ and $\lim g_n(x)\neq 0$ but $\int_{0}^{1} g_n\to 0$

Produce a sequence $(g_n):g_n(x)\ge 0,\,\forall x\in [0,1],\,\forall n\in\Bbb N$ and $\lim g_n(x)\neq 0,\,\forall x\in [0,1]$ but $\int_{0}^{1} g_n\to 0$ Im in need to clarify that Im talking of ...
0
votes
0answers
20 views

How to calculate the line integral $\int_{\vec{\gamma_j}}\langle\vec{v_k},d\vec{x}\rangle$ for $j=1,2$ and $k=1,2,3$

Here's the integral again: $$\int_{\vec{\gamma_j}}\langle\vec{v_k},d\vec{x}\rangle$$ Here's what I know about $\vec{\gamma_1},\vec{\gamma_2}:[0,1]\rightarrow\mathbb{R^3}$ $$\vec{\gamma_1}(t) = \left(...
5
votes
1answer
93 views

finding the series $\sum_{n=1}^\infty \frac{x^n}{n!} \frac{1}{n}$

My goal is to solve this series $$S(x) = \sum_{n=1}^\infty \frac{x^n}{n!} \frac{1}{n}$$ I did took the derivative first w.r.t $x$ $$S'(x) = \sum_{n=1}^\infty \frac{x^{n-1}}{n!}$$ which I ...
0
votes
3answers
43 views

Proving a Definite Integral Inequality without Geometrical Intuition

I solved an integral inequality problem using geometrical methods. However, I just cannot satisfy with them and want a without-geometrical-intuition proof, and I couldn't find one. Proof the ...
0
votes
2answers
57 views

How to substitute in an integral

I need to solve this integral by substitution: $$ \int \frac{1}{(1+\sqrt x)^2}\,dx $$ I know the substitution should be: $$ u= \sqrt x$$ and so $$du=\frac{1}{2\sqrt x}$$ but i can't understand how ...
0
votes
1answer
42 views

Calculation of double integral

I am trying to solve this integral $$ \int_{20}^{21}\int_{20}^{25}\frac{1}{\sqrt{2\pi}ga_{m}}\exp\Big{(}-\frac{1}{2} \frac{(a_{m}-(ba_{f}+c))^{2}}{g^{2}a_m^{2}}\Big{)}da_{m}da_{f} $$ with b,c,g ...
0
votes
0answers
22 views

Evaluate an integral with hyperbolic functions

I am trying to evaluate, respectively simplify the following integral expression $$f(t):=\int_{a}^{t-a} \frac{(\cosh(x-a)-\cosh(a))^{i\tau-1/2}}{(\cosh(x)-\cosh(a) )^{i\tau+1/2}} dx, \quad t>a,$$ ...
-1
votes
1answer
51 views

How to implement twice MATLAB integral build-in function for numeric integration? [on hold]

Suppose we have a function $F(\lambda) = \int\limits_{\lambda}^1 f(x) dx$, where $f(x)$ has no formula for antiderivative. We can easily calculate it by means of build-in MATLAB functions. Let's use $...
1
vote
0answers
9 views

Mean continuity of gradient

Let $f:\mathbb R^n\longrightarrow R$ be a differentiable function, and suppose $\nabla f$ is bounded. Prove that $$\lim_{r\to 0}\frac{1}{\omega_n r^n}\int_{B_r(x)}[\nabla f(y)-\nabla f(x)] dy=0.$$ ...
0
votes
1answer
51 views

How does one compute this heavy integral?

The integral is $$\frac{1}{2\pi i}\int_\Gamma\frac{\exp(z^2-\cos(iz)-4)}{z-2}dz$$ where $\Gamma$ is the unit circle. Here's how I tried to parametrize it: $z=e^{i\theta}$ on $\theta\in [0, 2\pi]$, ...
3
votes
2answers
62 views

Volumes of Revolutions : Lord of the Rings

Question: The "Lord of the Rings" has a collection of solid gold rings for different-sizes fingers. The cross section of each ring is a segment of a circle radius $R$ as shown in the diagram below. ...
0
votes
1answer
38 views

Integration of Rational function contain even power of variable

Evaluation of $\displaystyle \int\frac{1}{1+x^6}dx$ $\bf{My\; Try::}$Let $$I = \int\frac{1}{1+x^6}dx = \int\frac{1}{(1+x^2)(x^4-x^2+1)}dx$$ Using Partial fraction , above Integral is very lengthy, ...
0
votes
5answers
45 views

Integration with limits and options.

I found this exercise in an old exam but I don't know how to attack it because is a limit of an integration and I don't know if the limit affects the process of the integral or it makes it easier. The ...
0
votes
0answers
12 views

Multidimensional integration help

Please confirm my understanding of a function I would like ultimately to plot: $C_{ij}(l)=\int^{\chi_{h}}_{0}W_{i}(\chi)W_{j}(\chi)\frac{P(k=\frac{l}{\chi};z)}{\chi^{2}}d\chi$ where the weights with ...
0
votes
3answers
76 views

Indefinite integral of $\int \frac{x+2}{\sqrt{4x-x^2}}$, is my solution needlessly complicated?

This took awhile and since I am hoping to be able to compute these quickly in an exam setting I am hoping for a simpler way. Using the substitution $x=\sin^2t$ gets me: $$ I/2=\int \frac{\sin^2t+2}{\...
3
votes
1answer
36 views

Is the following integral identity true or not? [on hold]

Is the following statement true or not?$$\int_{-\infty}^\infty xf(x)\,dx = \left. {d\over{dt}} \int_{-\infty}^\infty e^{tx}f(x)\,dx\right|_{t = 0}$$
1
vote
1answer
44 views

Integration of $\int \frac{1}{x^{1/3}(x^{1/3}-1)}dx$ [on hold]

Integrate the following function $$\int \frac{1}{x^{1/3}(x^{1/3}-1)}dx$$ Could someone give me slight hint to solve this question?
0
votes
3answers
46 views

How to use integrals to evaluate a function?

Textbooks often show: $f(x)=\int f'(x)\ dx$ But how do we evaluate $f(x)$ at a specific x, let's say $f(2)$? Let's say $f(x)=x^2 ,\ f'(x) = 2x$ then $f(3) = 9$ but on the right hand side, ...
2
votes
2answers
36 views

MVT for integrals: strict inequality not needed before applying IVT?

I've looked at Nigel Overmars's answer here: http://math.stackexchange.com/a/630429/349828 His proof is essentially identical to the one I wrote myself and to the one given by my analysis professor ...
1
vote
0answers
15 views

Examples where ostragodsky's method is needed for integrating rational functions

I found out about Ostragodsky's method for integrating rational functions and thought it was pretty cool. However, I have never encountered any examples where it seemed needed (rather than just ...
1
vote
0answers
41 views

Double Gaussian Integral

I am interested in the following integral $$\int_{-a}^adx\int_{-a}^a\mathop{\mathrm{d}y}xy\frac{1}{\sqrt{b}}\exp\left[-\frac{(x-y)^2}{2b}\right]\sqrt{\frac{2}{c}}\exp\left[-\frac{(x+y)^2}{4c}\right].$...
0
votes
1answer
48 views

Counter Example for Limit of $\|f\|_p$ in infinity convergence, When Measure space is not finite [on hold]

I found a proof for this fact that limit of $\|f\|_p$ when $p \to \infty $ is $\|f\|_{\infty}$ in here when $f:X \to R $ and $X \in L^p$ measure space is finite. But I need a counter example for ...
5
votes
4answers
108 views

Tips for integrating $\int \frac{dx}{1+\cos(x)}$

I tried the following $$ \int \frac{dx}{1+\cos(x)}=\int \frac{1-\cos(x)}{1-\cos^2(x)}\,dx= \int \frac{1-\cos(x)}{\sin^2(x)}\,dx\\ =\int \frac{1}{\sin^2(x)}\,dx-\int \frac{\cos(x)}{\sin^2(x)}\,dx=\int ...
1
vote
1answer
24 views

Question on integral computation

I am trying to actually compute $\int \sqrt{\cosh{y}-\cos{x}}e^{inx} dx$ . Would someone please provide me a mathematical approach into solving $\int \sqrt{2-\cos{x}}e^{inx} dx$ and i shall figure ...
5
votes
1answer
108 views

How do I find a closed form of ${\pi^{2n}\over \zeta(2n)}\int_{-1}^{1}{x^{2n-2}\over \pi^2+(2\tanh^{-1}{x})^2}dx$?

How do I evaluate the closed form for $g(n)$? Where n is an integer, $n\ge 1$ $${\pi^{2n}\over \zeta(2n)}\int_{-1}^{1}{x^{2n-2}\over \pi^2+(2\tanh^{-1}{x})^2}dx=g(n)$$ Make a subsititution $u=\...
3
votes
0answers
65 views

Integrating an inverse sum of roots

I recently thought about this question, and I attempted to solve it $$I_n = \int_0^1 \frac{1}{1+x+\sqrt{x} + \cdots + \sqrt[n]{x}} \mathop{ dx}$$ for $n\ge 2$. I've made next to no progress, although ...
1
vote
1answer
62 views

Close form solution for $\int_{0}^{\infty}\frac{x^\alpha}{e^x-1}$ [on hold]

Is there the close form solution for the integral is given by $$\int_{0}^{\infty}\frac{x^\alpha}{e^x-1}\,dx$$ where $\alpha>0$