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
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5answers
100 views

Integrate $4x/(x^4-1)$ dx

I'm having trouble solving the integral $$\int_{5/4}^{13/12}\frac{4x}{x^4-1}\,dx$$ I have a feeling it is to do with the log integration identity but can't seem to manipulate it without involving ...
1
vote
1answer
31 views

How would I reverse the order of these integrals?

In the question, x is the first integral at the moment. Its $\sin^{-1}(y^2)<x<\pi/2$. Y is the second integral which needs to become the first. Its $0<y<1$. I know how to integrate the ...
0
votes
0answers
39 views

Different forms of remainder in Taylor series

In the literature, it is common to find different forms of the remainder function in a truncated Taylor series. To name a few: Integral form Lagrange form Cauchy form First, can you tell me any ...
4
votes
1answer
63 views

Calculate $\int_0^1 f(x) dx$, where $f(x)= \begin{cases} 0 & x= x\in C \\ \frac{2}{7^n} & x= x\not \in C \end{cases} $, with $C$ the Cantor set.

I am looking back at notes, and problems from the semester, and I came across this problem that I am having trouble solving. Let $$f(x)= \begin{cases} 0 & x= x\in C \\ \dfrac{2}{7^n} ...
2
votes
0answers
30 views

antiderivative of $\psi'(u)$ for $u\in W^{1,2}_0((a,b))$

Let $u\in W^{1,2}_0((a,b))$ and $\psi'$ the derivative of a convex function $\Psi\in C^1(\mathbb{R})$. If I want to consider the antiderivative of $\psi'(u)$, what happens with the $u$ inside $\Psi(u)$...
1
vote
2answers
60 views

If $\ln x$ is integrable, then is $x \ln x$ also integrable?

I have a very simple problem. Assume we have a finite measure $\mu$ on $[1,\infty)$, and \begin{align} \int_1^\infty t ~d\mu(t) < \infty. \end{align} My question is if this implies \begin{align} \...
1
vote
1answer
80 views

Compute $\lim_{n\to\infty}\int_0^1\frac{x\sin{nx}}{1+x^2n^6}dx$

$\lim_{n\to\infty}\int_0^1\frac{x\sin{nx}}{1+x^2n^6}dx$ My Work: By the comparison principle: $$\lim_{n\to\infty}\int_0^1\frac{x\sin{nx}}{1+x^2n^6}dx\le\lim_{n\to\infty}\int_0^1\frac x{1+x^2n^6}dx$$...
1
vote
0answers
79 views

Fundamental Theorem of Calculus application question

I have a question that I believe I have the solution to, but I am not sure and there is no way of checking. As such I was hoping someone could look over my solutions and provide me with some feedback. ...
0
votes
0answers
24 views

Does this integral of Appell F_1 converge?

I'm interested in whether or not integrals of the form $$\int_{0}^{1}\mu^{\alpha}F_{1}\left(\frac{\alpha}{2};1,-1;\frac{\alpha+2}{2};\mu^{2},-\beta\mu^{2}\right)\mathrm{d}\mu$$ converge, and if so ...
5
votes
3answers
469 views

The value of double integral $\int _0^1\int _0^{\frac{1}{x}}\frac{x}{1+y^2}\:dx\,dy$?

Given double integral is : $$\int _0^1\int _0^{\frac{1}{x}}\frac{x}{1+y^2}\:dx\,dy$$ My attempt : We can't solve since variable $x$ can't remove by limits, but if we change order of integration, ...
1
vote
2answers
76 views

Solving this rather tedious integral

I need help solving this integral $$\left \langle x \right \rangle = \frac{2}{a}\int_{0}^{a}x\sin^{2}\left ( \frac{m \pi x}{a} \right )\,dx$$. I have tried to reduce $$\sin^{2}\left ( \frac{m \pi x}{...
4
votes
2answers
80 views

Lipshitz Integral for $a=0$

I knew that this, $$\displaystyle{\int_0^\infty e^{-ax}J_0(bx)dx=\frac{1}{\sqrt{a^2+b^2}}},$$ holds for $a>0$ but, in an exercise from Arfken, it said that this holds for $a\geq0$. How can I prove ...
0
votes
3answers
38 views

Using Divergence Theorem to evaluate the integral

Find the value of $$ \iint_{\Sigma} <x, y^3, -z>. d\vec{S} $$ where $ \Sigma $ is the sphere $ x^2 + y^2 + z^2 = 1 $ oriented outward by using the divergence theorem. So I calculate $ div\vec{F}...
0
votes
1answer
74 views

Determine poles and residues of contour integral using Laurent series

I want to find the residues of the integral $F = \int_{-\infty}^{\infty} \dfrac{1}{x+(a-ib)} \dfrac{1}{\exp(-x/c)-1} dx$ I know that $x=-(a-ib)$ is a simple pole which contributes a non-zero residue....
1
vote
2answers
38 views

Taking the derivative of an integral using chain rule

$2 \frac d {dy} (\int_0^{\sqrt y}3x^2 dx) $ I know that this gives you $3y^{\frac 1 2}$ as a result, if done step by step, but I've been told I can use chain rule to to do it in a single step. I've ...
0
votes
1answer
61 views

How can one derive the circumference of a circle using integrals?

Many proofs for the area of a circle start with something like $$ A(r) = \int_0^r 2 \pi t dt $$ such as at https://en.wikipedia.org/wiki/Area_of_a_disk#Onion_proof , but I don't understand how to ...
4
votes
2answers
43 views

Integrate $\cos(x^3)$ over given bounds

I'm not really sure at all how to integrate this function. I was wondering if anyone could help me out, $$\int^4_0 \int^2_{\sqrt y} \cos(x^3) \, dx \, dy$$ The answer choices are ${1 \over 3}\sin(...
1
vote
1answer
29 views

Convergence of Definite Improper Integrals of the Form $1/x$

Given a simple integral of the form: $$ \int ^1 _{-1} \frac{1}{x} \, dx =\lim _{a\rightarrow 0} \int ^1 _a \frac{1}{x} \, dx + \int ^a _{-1} \frac{1}{x} \, dx$$ Is it possible to say that this ...
0
votes
1answer
41 views

Initial value problem without explicit constant finding

So I was reading my professor's differential equation book until I came across a weird way of calculating a initial value problem without explicity having to calculate the constant that made that ...
2
votes
1answer
40 views

Use The Divergence Theorem to evaluate the flux

Let $ \vec{F}(x, y, z) = (\sin z + xy^2)\vec{i} + x^2e^{3z}\vec{j} + (\cos ^3x + x^2z)\vec{k}. $ Let $ T $ be the surface bounding the region $ R $ given by $ x^2 + y^2 \leq z \leq 6 - \sqrt{x^2 + y^...
0
votes
2answers
48 views

Newtons Law of Cooling (Integration)

The question is : An apple crumble is taken out of the oven at 7:30pm. At that time, its temperature is 100 degrees Celsius. At 7:40pm, ten minutes later, the temperature of the apple is 80 ...
1
vote
1answer
25 views

Triple integrals, find the boundaries

I have this triple integrals and before I can grind it down I have to find the boundaries. But I can't figure out how to find the boundaries for $x$ I know the boundaries are: $0<x<2$ $x<...
0
votes
0answers
36 views

Is there any integral for sin(x)/x [duplicate]

What is the integration of the function $$f(x)= \frac{\sin(x)}{x}$$ I have seen many stopping at here.. If I integrate using integration by parts it goes on..
0
votes
0answers
31 views

Inverse Substitution ( Trigonometric Integration Rule Generalized )

I read that we can generalize the kind of substitution that is used in Trigonometric Substitution. The idea is that we replace the old variable with a new one (like we do in U-Substitution) but unlike ...
0
votes
2answers
25 views

Problem using triple integrals and the directional derivative

Sorry for the vague question, but I'm so lost on this problem I don't even really know what it's asking of me. If anyone could show me how to do it I would appreciate it so much. All I can think of is ...
1
vote
0answers
105 views

Can be justified $\zeta(3)=\lim_{n\to\infty}-3\sum_{k=1}^n\sum_{\nu=0}^\infty\frac{(-1)^{\nu+1}}{\nu+1}\binom{3k^3-1}{\nu}$?

My main goal is understand useful facts about my computations, that could be wrong, the way shold be too a street without exit looking for a evaluation of Apéry's constant, since I don't use any ...
4
votes
1answer
45 views

Prove that $\int_a^b \left( \int_c^d f(x,y)dy\right) dx=\int_c^d \left( \int_a^b f(x,y)dx\right) dy$

Let $f$ be continuous function on $[a,b]\times [c,d]$. Prove that Prove that $$\int_a^b \left( \int_c^d f(x,y)dy\right) dx=\int_c^d \left( \int_a^b f(x,y)dx\right) dy$$ First of all, note that ...
2
votes
2answers
39 views

Why does $\int_{-L}^{L} \sum_{n=1}^{\infty}a_n\cos \frac{n\pi x}{L}=\sum_{n=1}^{\infty}a_n\int_{-L}^{L}\cos \frac{n\pi x}{L}$

Why does $$\int_{-L}^{L} \sum_{n=1}^{\infty}a_n\cos \frac{n\pi x}{L}=\sum_{n=1}^{\infty}a_n\int_{-L}^{L}\cos \frac{n\pi x}{L}$$ This is used in a derivation of the Fourier coefficients. I see why ...
0
votes
0answers
24 views

integrate the improper function in (0,1)

Find $\int_{0}^{1} f(x)\mathrm{d}x$ hwere $f(x) = 2x\sin(\frac{1}{x})-\cos(\frac{1}{x})$, $x$ in $(0,1)$ 0, for all other values
1
vote
0answers
30 views

Proof of this integral identity

I was going through some old papers on CFT particularly, 'CONFORMAL COVARIANT CORRELATION FUNCTIONS - Ferrara and Parisi' where an integral identity is used to motivate shadow operator formalism in ...
3
votes
2answers
69 views

Green's Theorem on Line Integral

I am asked to find the line integral for the following field: $$F = (e^{y^2}-2y)i + (2xye^{y^2}+\sin(y^2))j$$ On the line segment with points $(0,0),(1,2)$ and $(3,0)$. I have to do it with Greens ...
2
votes
3answers
122 views

Evaluating the flow rate of water across the net

A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by $\vec{v}=(x-y)\vec{i}+(z+y)\vec{j}+z^2\vec{k}$ and the net is ...
1
vote
0answers
21 views

Changing the measure

Suppose that I have the following situation: everything takes place on complex plane, I have some probabilistic measure $\mu$ and suppose that I have a family of function $(u_w)_w$ parametrised by $w$ ...
0
votes
1answer
27 views

Volume between $z = 3\sqrt{x^{2} + y^{2}}$ and $x^{2} + (y-1)^{2} = 1$ and $z = 0$

Find the volume between $z = 3\sqrt{x^{2} + y^{2}}$ and $x^{2} + (y-1)^{2} = 1$ and $z = 0$ I am not sure how to approach finding the limits of integration. Would I need to change coordinate systems?...
1
vote
3answers
60 views

Why change the sign of the integral when switching the limits of integration?

So I'm refreshing on integration and venture across a property of definite integrals I've been taking for granted for quite some time: \begin{equation}\int_{a}^{b} f(x) \mathrm{d}x =-\int_{b}^{a} f(x)...
1
vote
1answer
60 views

how to integrate the definite integral using residue theorem? [duplicate]

How to evaluate $\int_{0}^{\infty}\dfrac{1}{x^a+1}dx$, where $a>1$. I don't know where to start since $x^a+1$ could have infinitely many roots, then it is impossible(?) to evaluate its residues. ...
3
votes
2answers
53 views

Prove that there is some points $x_0$ in the interval $[a,b]$ at which $f(x_0)=0$.

Suppose that the continuous function $f:[a,b]\rightarrow\mathbb{R}$ has $\int_{a}^{b}f(x)=0$. Prove that there is some points $x_0$ in the interval $[a,b]$ at which $f(x_0)=0$. Proof: Since $\int_{...
2
votes
3answers
68 views

Question about varying density of a sphere to find its mass.

I have a question about the process to find the mass of a sphere with a varying radial density in respect to the radius. It's something really simple, but I would like someone to explain it me. Say ...
2
votes
2answers
43 views

Integrability of $\frac{1}{|x|^d}$

For $x \in \mathfrak R^d$,why is $\int_\limits{\{x; |x|\geq 1\}} \frac{1}{|x|^d} dx = \infty$ in Lebesgue integral? It's hinted to apply Tonelli Theorem (Fubini Theorem) and use the fact that $\frac ...
2
votes
0answers
43 views

Integral of the product of a power function and an arbitrary exponentiated function

I was only able to find integral tables that solve $$f(t)=\int t^c e^{kt}dt$$ but the integral I'm trying to solve has a function, not a constant, for the exponent: $$f(t)=\int t^c e^{g(t)}dt$$ Is ...
0
votes
1answer
200 views

Derivatives and Integrals of Polynomials and more.

I noticed that if I had a function $f(x)=x^n$ where $n$ is an integer, then $\lim_{m\to{n^+}}f^{(m)}(x)=n!$ where $f^{(m)}(x)$ is the $m$-th derivative. Also, $$\lim_{m\to{n^-}}f^{(m)}(x)=\frac{(-1)^...
0
votes
0answers
30 views

Find volume of the region

Find the volume of the region when B is the region bounded by the cylinder $$x^2 + 3z^2 = 9$$ and the planes $$y = 0$$ and $$x + y = 3$$. $$ \int_{-3}^{3} \int_{0}^{3-x}\int_{-\sqrt{\frac{9-x^2}3}}^{\...
2
votes
1answer
41 views

A function and its derivative chasing tails

For which $t\ge0$ does there exist a differentiable function $f$ with $f(0)=0$, $f'(x)>f(x)$ for all $x>0$ and with $f'(0)=t$? This question was inspired by (and is a variation of) the ...
-1
votes
1answer
29 views

Find surface area of the portion of the planar surface $S(u,v)=<u,v,1-u-v>$

Where $(u,v)$ belong to the triangle in the $uv$-plane with vertices $(0,0) (1,0) (0,1)$. I am completely stuck here!! I am sure I will use double integrals (or at least I think so) and also the ...
8
votes
3answers
136 views

how to integrate this $\int_0^{\infty} r^2 e^{\frac{-r^2}{2}} \, dr$?

What am I doing wrong when integrating this? $$\int_0^{\infty} r^2 e^{\frac{-r^2}{2}} \, dr$$ I used integration by parts and set $u=r^2$ and $dv=e^{\frac{-r^2}{2}}dr$ and I get $$-re^{\frac{-r^2}{2}}...
1
vote
1answer
30 views

Determine equation linking 2 variables

I have some data with 3 variables P, s and t. I know that P is related to s as per the first equation below. I also know that s is equal to the area under a graph of P on the y-axis and t on the x-...
0
votes
1answer
62 views

Multiple integral. How Dirac delta change limits if they are finctions?

Good evening! During my calculation I meet such integral: $$\int\limits_{0}^{1} dx \int\limits_{f(x)}^{g(x)} dy \,\delta(x - y)$$ I know how Dirac delta act on functions which we integrate, but I ...
0
votes
1answer
28 views

Lower Reimann integral

I am reading Elements of Integration by Bartle and I came across this. "If f is a bounded function defined on an interval [a,b] and if f is not too discontinuous .......In particular the lower ...
10
votes
1answer
158 views

Is $\int_{M_{n}(\mathbb{R})} e^{-A^{2}}d\mu$ a convergent integral?

Is the following integral a convergent integral? Can we compute it, precisely? $$\int_{M_{n}(\mathbb{R})} e^{-A^{2}}d\mu $$ Here $\mu$ is the usual measure of $M_{n}(\mathbb{R})\simeq \mathbb{R}^{n^...
2
votes
2answers
45 views

Integral $\int^\ell_{-\ell} r/\sqrt{L^2+r^2}^{\,3} \, dL$

We tried to solve a magnetics problem and ended up with $$\int^\ell_{-\ell} \frac{r}{\sqrt{L^2+r^2}^{\,3}} \, dL.$$ How do I solve this integral?