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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1answer
27 views

Integrals with $e^z$ [on hold]

I need help starting this question. Calculate the integral of $$f(x,y,z)=e^z$$ over the portion of the plane $$x+2y+z=4$$ where $x,y,z$ is greater than or equal to 0.
0
votes
1answer
36 views

Problem with a limit with a integral in it

Suppose that the temperature in a long thin rod placed along the $x$-axis is initially $\frac{C}{2a}$ if $|x| \leq a$ and $0$ if $|x| > a$. It can be shown that if the heat diffusivity of the rod ...
0
votes
1answer
13 views

Triple Integral of 6xy dV, why is lower bound of z 0 instead of 1?

In the Stewart Calculus Textbook 7th Edition, problem 13 in chapter 15.7 states: "Evaluate the triple integral: Triple Integral within E of 6xy dV, where E lies under the plane z=1+x+y and above the ...
1
vote
1answer
45 views

How can I calculate $\int_1^2 \int_{\sqrt{x}}^x \sin\left(\frac{\pi x}{2y}\right) \,dy \, dx$?

Good night i have problem solving this integral. $$\int_1^2 \int_{\sqrt{x}}^x \sin\left(\frac{\pi x}{2y}\right) \,dy \, dx$$ I make the area of integration, but i cannot solve the integrat, i don't ...
0
votes
0answers
22 views

Name or reference about a inequality with integrals?

I have wrote down some class notes and I think I copied something wrong. It is an integral inequality; $$\iiint_{B^n}|\nabla\psi|^2\frac{1}{|x|^{n-2}}dV\leq C\iint_{\partial B^n}|\psi|^2dA$$ where ...
1
vote
0answers
36 views

the sum-int exchange

Why is the sum-int exchange allowed in the following equality ($f$ is a $C^1$ function): $$ \sum_{m\geq 1} \int_0^{\pi}f'(t) \frac{\sin((2m+1)t)}{2m+1} dt = \int_0^{\pi}f'(t) \sum_{m\geq ...
0
votes
1answer
31 views

The reason of $\int_{-\infty}^{\infty}\mu_k^2(x)dx=1$

Is there anyone could tell me why if $$\sum_{k \geq 0} e^{it \sqrt{-\lambda_k}}=\int_{-\infty}^{\infty} (\sum_{k \geq 0} e^{it \sqrt{-\lambda_k}} \mu_k^2(x))dx= \sum_{k \geq 0} e^{it ...
0
votes
2answers
36 views

Relation between $\int_{a}^{b} f(x) dx$ and $\int_{a}^{b} (1-f(x)) dx$

Say you're expected to work out $\int_{0}^{\pi/3} \sin^2(x) dx$ solely from the result $\int_{0}^{\pi/3} \cos^2(x) dx$. It can be transformed into $\int_{0}^{\pi/3} (1-\cos^2(x)) dx$, but then what?
12
votes
1answer
311 views
+100

Why does the hard-looking integral $\int_{0}^{\infty}\frac{x\sin^2(x)}{\cosh(x)+\cos(x)}dx=1$?

I have to ask this question; most looking complicated definite integral yield not so nice closed form or irrational numbers or mixed of what ever ect. Why is this particular hard looking integral ...
5
votes
3answers
101 views

Solution of integral $\int \frac{\sin (x)}{\sin (5x) \sin (3x)} dx$

Find the following integral $$\int \frac{\sin (x)}{\sin (5x) \sin (3x)} dx$$ I don't know how to deal with the $\sin (x)$ in the numerator. If it had been $\sin (2x)$ then we could have used $\sin ...
0
votes
2answers
26 views

Generating definite integral for unit circle without consulting tables

I realize that I can consult integration tables for integrals like this, but I wanted to test my knowledge of integration techniques to solve this integral for the area of the unit circle. ...
1
vote
2answers
22 views

expressing contour integral in different form

Hi I have a short question regarding contour integration: Given that $f(z)$ is a continuous function over a rectifiable contour $z = x + iy$. If $f(z) = u(x,y) + iv(x,y)$, why does it follow that the ...
0
votes
0answers
21 views

Integral of convex function applied on a function

Let $f$ be an integrable function of $\mathcal{L}(\mathbb{C},\mathbb{R})$, measure Lebesgue. I want to prove that there exists an increasing convex function $H:\mathbb{R}^+\rightarrow\mathbb{R}^+$ ...
0
votes
1answer
30 views

Definite Gaussian/exponential integral

I've been revising some quantum mechanics, and I was wondering how I would calculate a specific standard deviation. The wave function I am working with is ...
-2
votes
0answers
24 views

integration of a non-negative continous functions

Let $f:[0,1] -> \mathbb R$ be continuous such that $f(t) \geq 0$ for all $t$ in $[0,1]$. Define $g(x) = \int_{0}^{x}f(t)dt$ then a) $g$ is monotone and bounded. b) $g$ is monotone, but not ...
0
votes
0answers
21 views

Integral computation with Mathematica and Sympy differ

To compute the integral: $I = \int_{0}^{+oo} ue^{Au^{2}+Bu}du$ where $A<0$ and $B>0$ I have tried both Mathematica and Sympy but they yield different results: Mathematica yields: $ I = ...
0
votes
0answers
80 views

If $ \int_{0}^{2}\frac{ax+b}{(x^2+5x+6)^2}dx = \frac{7}{30}\;,$ Then value of $a^2+b^2$

If the value of Definite Integral $\displaystyle \int_{0}^{2}\frac{ax+b}{(x^2+5x+6)^2}dx = \frac{7}{30}\;,$ Then value of $a^2+b^2$ $\bf{My\; Try::}$ We can write it as ...
1
vote
2answers
30 views

Simplification idea for finding antiderivative

Is there a simple way of finding the anti-derivative $F$ (i.e. $F(x)=\int f(x)dx$) of $$f(x)=\frac{1}{(\sqrt{x}-1)\sqrt{x}}$$ I've managed to do it by 2 by parts integrations in row, but that took ...
3
votes
2answers
62 views

If $f(x) = ax^2+bx+c$ and $f(0) = 0$ and $f(2) = 2\;,$ Then minimum value of $\int_{0}^{2}|f'(x)|dx$

Consider the polynomial $f(x) = ax^2+bx+c$ and $f(0) = 0$ and $f(2) = 2\;,$ Then Minimum value of $\displaystyle \int_{0}^{2}|f'(x)|dx$ $\bf{My\; Try::}$ From $f(0) =0\;,$ We get $c=0$ and ...
2
votes
1answer
45 views

Where is my mistake calculating $\int_{-\infty}^{\infty}\frac{x\sin(x)}{x^2+4}~dx$?

Where is my mistake calculating $$\int_{-\infty}^{\infty}\frac{x\sin(x)}{x^2+4}~\text{d}x$$ Let $$f(z)=\frac{z\sin(z)}{z^2+4}$$ it has simple poles at $\pm 2i$. We take the standard half circle path ...
2
votes
3answers
99 views

Real Analysis question on FTC, Integral

Let $g:[0,1] \rightarrow \mathbb R$ be a continuous function and assume that $$ \int_{0}^{1} g(x) \phi'(x) dx = 0 $$ for all continuously differentiable functions $\phi: [0,1] \rightarrow ...
1
vote
0answers
24 views

Use double integrals to find the volume of the solid obtained by the rotation of the region

I am unsure how to set up the integrals. The integration should be done in polar coordinates. Here is what is give: $D = \{ (x,y,z) | x^2 \le z \le 6 - x, 0\le x \le 2, y = 0\}$ in the xz-plane ...
0
votes
2answers
74 views

Is $\frac{1}{x^2}$ Lebesgue integrable while $\frac{1}{x}$ is not?

My textbook defined integrability as $f$ is said to be Lebesgue integrable if $\int{}f$ is finite. I heard that $\frac1x$ is not Lebesgue integrable, but $\frac{1}{x^2}$ is Lebesgue integrable. I do ...
4
votes
1answer
58 views

How do you show that $\int_{0}^1\frac{dx}{x^x}=\sum_{k=1}^\infty\frac{1}{k^k}$?

My task is this: i) Find the sum to$$1-x\ln x +\frac{1}{2}(x\ln x)^2-\ldots+\frac{(-1)^k}{k!}(x\ln x)^k+\ldots$$ (ii) The great norwegian mathematician Atle Selberg showed that ...
1
vote
1answer
46 views

Limit of a Riemann sum: $\lim_{n\to\infty} {n^5 \sum^n_{r=0}\frac1{(n^2+r^2)^3}} $

Required to find $\lim_{n\to{\infty}} {n^5 \sum^n_{r=0}\frac{1}{(n^2+r^2)^3}} $ $\lim_{n\to{\infty}} \frac{1}{n} \sum^n_{r=0}(\frac{n^2}{n^2+r^2})^3$ $\lim_{n\to{\infty}} \frac{1}{n} ...
0
votes
0answers
34 views

How to integrate $\cos2\pi\left(x+\frac{n}{x}\right)$

This is a follow up question of Integrate $\cos^2(\pi x)\cos^2(\frac{n\pi}{x})$. By using product to sum formula, this could be converted to question to integrate ...
1
vote
1answer
46 views

Estimation of Integral $E(x)$

How can we prove $$\frac{1}{2}e^{-x}\ln\left(1+\frac{2}{x}\right)<\int_{x}^{\infty}\frac{e^{-t}}{t}dx<e^{-x}\ln\left(1+\frac{1}{x}\right)\;, x>0$$ $\bf{My\; Try::}$ Let $\displaystyle ...
0
votes
0answers
10 views

Can quaternions be used to represent rotation rate?

A quaternion is a useful tool for representing a rotation, or change in attitude. If a quaternion $q$ represents a rotation, and $v$ a vector, then $v'=qvq^*$ rotates the vector, where the multiply ...
0
votes
1answer
27 views

Problem solving a partial derivative with a integral. [on hold]

Good night, i have a serious problem solving this partial derivative: $f(x,y)=\int_{y}^{x}e^{t^{2}}dt$ I don't know how i can start this, please give me a help, don't do it the exercise, only explain ...
8
votes
3answers
145 views

Need help with $\int_{-\infty}^\infty \frac{x^2 \, dx}{x^4+2a^2x^2+b^4}$

I'm having trouble trying to evaluate this definite integral. Mathematica didn't help much. $$\int_{-\infty}^\infty \frac{x^2 \, dx}{x^4+2a^2x^2+b^4}$$ where $a$, $b$ $\in \Bbb R^+$. Is it possible ...
0
votes
3answers
36 views

Surface integral of a scalar over a unit cube.

Evaluate the following integral $$\iint_S (x+y+z) \, dS$$ where $S$ is the surface of the cube $[0,1] \times [0,1] \times [0,1]$ Honestly, I don't know what to do. All I know is that you have to ...
1
vote
1answer
33 views

Kernel and Image of an integral.

Im struggling to answer a question where $F: P_{2}(\mathbb{R}) \rightarrow P_{3}(\mathbb{R}) $ $$F(f)(x)=\int^{x+1}_{2-x} (1-t)f(t) dt$$ So to find the Kernel do i set the integral equal to 0 and ...
1
vote
1answer
92 views

Show $R(x)=o(x^3)$

I got $$R(x)=4! \, x^4 \int _0^{\infty} \frac{1}{(1+xt)^{5}}e^{-t} \, \, dt$$ is this correct? I have no idea what to do for the last part of ii
2
votes
1answer
46 views

Magnetic field by current in an infinite cylinder

Let $V\subset\mathbb{R}^3$ be an infinitely high solid cylinder of radius $R$, with its axis coinciding with the $z$ axis, entirely enclosed by the cylinder's lateral surface. Then, for any constant ...
3
votes
2answers
53 views

Find The Volume of the solid in the first octant , limit by: $ x^2+y^2=4 $ and $z+y=3$

Find the volume of the solid in the first octant , limit by: $ x^2+y^2=4 $ and $z+y=3$. $x$ and $y$ range from $0$ to $2$. $$\int_0^2 \int_0^2 y-3 \,dy\,dx $$ is correct?
0
votes
2answers
65 views

help for an integral

I need help calculating this integral: $$\int_0^x \frac{2(e^{\gamma u}-1)}{(\gamma+\kappa)(e^{\gamma u}-1)+2\gamma} du$$ I tried with the integration by parts but the situation seems to get ...
1
vote
0answers
49 views

unbounded solution, lim inf of trace,

Show that if $\lim \inf_{t\rightarrow \infty} \int_{t_0}^t \operatorname{tr}\left(A(s)\right)ds= \infty $ then the linear first-order system $x'(t)=A(t)x(t)$ where $A \in C\left(I, \mathbb{R}^{n\times ...
0
votes
0answers
59 views

Trivial equation but with integral

Let's consider the following equation: $$\int_{a}^{x} f(t)dt = K$$ where $a, K \in \mathbb{R}$. Suppose that $a, K$ and the function $f$ are known and that the equation should be used to determine ...
-3
votes
0answers
40 views

Please help me with these 4 questions, thanks. [on hold]

This is the image with the questions: (Large version)
0
votes
2answers
43 views

Integration substituion

I need to use a suitable substitution to show $$\int {dx \over \sqrt{a^2+x^2}} = \text{arcsinh} \left(\frac{x}{a}\right)+C$$ but I am not sure what substitution to use. Any help would be great. ...
-1
votes
3answers
95 views

find the value $\int_{0}^{+\infty}\left(\frac{x^2}{e^x-1}\right)^2dx$ [on hold]

Find the $$I=\int_{0}^{+\infty}\left(\dfrac{x^2}{e^x-1}\right)^2dx$$ Let $x=\ln{t}$,$$I=\int_{1}^{+\infty}\dfrac{\ln^4{t}}{t(t-1)^2}dt$$
0
votes
0answers
17 views

Integral of implicit function - geometric meaning [on hold]

What is the geometric meaning of implicit function $$f(x,y) = 0$$ integral? Is it the same as for explicit function, eg. area, volume etc.. and we are computing it for the $0$ result, or is there ...
0
votes
1answer
18 views

“Nonlinear cosine” integral

Let $\alpha > 1$, $\xi \in\mathbb{R}$. and $\chi_A$ be the characteristic function of the set $A$. Are there some known ways of computing (or estimating in terms of $\xi$) of this kind of ...
-3
votes
1answer
53 views

Double integral involving $y /\sin y$ [on hold]

Let $f\colon \Bbb R^2\to \Bbb R$ be defined by $${f(x,y)= \begin{cases} \frac y{\sin y}, &y\neq 0\\ 1, &y=0 \end{cases}}$$ Then the integral $${\frac ...
0
votes
1answer
40 views

Numerical integration in Matlab (Gaussian 3 point quadrature)

Write a Matlab function that applies the Gauss three point rule to N sub-intervals of $[a, b].$ The input parameters should be the name of the function being integrated, $a, b,$ and $N$. Attempt: ...
1
vote
2answers
37 views

Am I interpreting antiderivatives the right way?

Here the antiderivative of $f(x)=x^2$ is $F(x) = x^3/3 + C$. If the constant of integration is $C = 0$, then for any $x$, $F(x)$ would give me the area under the curve of $f(x)$ from $0$ till $x$ ...
0
votes
1answer
25 views

How can I understand $x(b)=x(a)+\int_a^{b}f(s,x(s))\,ds$?

I am trying to understand this integral form of the ordinary differential equation: $$x(b)=x(a)+\int_a^{b}f(s,x(s))\,ds\quad\text{for }a\leq t\leq b$$ I tried to pick a concrete example: Let ...
0
votes
1answer
38 views

Double Integral Range

we want the Surface between the two equation: $r= 1 + \cos\theta$ $r = 1$ (circle) we can use a duble integral to solve this: $$ S = \int_{-\pi/2}^{\pi/2} \int_1^{1+\cos\theta} r \,dr\,d\theta ...
2
votes
1answer
74 views

TIFR GS 2015 computer science: $G = \lim_{n\to\infty}(n+1)\int_{0}^{1} x^{n} f(x) dx$

Following expression was asked to be evaluated in TIFR GS 2015 exam, $$G = \lim_{n\to\infty}(n+1)\int_{0}^{1} x^{n} f(x) dx$$ where $x \in [0, 1]$ and $f(x)$ be any real valued continuous function. ...
1
vote
1answer
56 views

Evaluate the indefinite integral $\int \frac{t\sin at}{b^2+t^2}dt$

It is known DLMF (25.2.8) that for $\Re s>0$ and for integers $N\geq 1$ $$\zeta(s)=\sum_{k=1}^N\frac{1}{k^s}+\frac{N^{1-s}}{s-1}-s\int_{N}^\infty \frac{x-\lfloor x \rfloor}{x^{s+1}} dx,$$ where ...