Tagged Questions

All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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4 views

Double Integral Mistake with Parametric Equation

I'm trying to figure out the mass of an object bounded by $y=0$ and $y=\sqrt{1-x^2}$ the density at a given point is proportional to its distance from the origin; $\rho(x,y) = kxy$. So I set it up ...
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1answer
30 views

Is there a function whose definite integrals are all 0?

Is there a continuous function $f: [0,1] \rightarrow \mathbb{R}$ such that $f(x) \neq 0$ for some $x \in [0,1]$ and, if we define $F_n(x) = \int_{0} ^ {x} F_{n-1}(t) dt $ (where $F_0(x)=f(x)$), then ...
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14 views

Numerically solve integral with a function as variable of integration

I want to use a function as variable of integration for example in evaluating the integral: $\int_0^1 e^{\cos x}f(\sin x)d\cos x$ in which $f(x)$ is an arbitrary function.
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14 views

Integeration of 2nd derivative

I have a question regarding solving the integral of the partial. Here is the equation: $$R_m = \int \psi^m \frac{\partial^2}{\partial\psi^2} \left[\left\langle \epsilon_\phi | \psi \right\rangle ...
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14 views

Integrability vs. Estimate

Given a finite measure space $\mu:\Sigma\to\mathbb{R}_+$. Consider measurable functions $f:\Omega\to\mathbb{C}$ and $g:\Omega\to\mathbb{C}$. Then the equivalence holds: ...
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1answer
34 views

Why am I obtaining an imaginary part for my integration

I try to solve an integration as follows, $$\int \frac{sy^{-1}}{(1+sy^{-1})} \text{exp}(-\sqrt{y})dy$$ as you can see its pretty complicated, and I get an answer like the following using Wolfram ...
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0answers
9 views

Specific Type of Dominated Convergence (Spectral Measures)

Reference See Birman and Solomjak, Spectral Theory of Self-Adjoint Operators in Hilbert Space, chapter 5 subparagraph 4.1, page 133... Question It is introduced a specific type of convergence, ...
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2answers
18 views

Why is $\vec{s}=\frac{\vec{r}}{V^\frac{1}{3}} \Leftrightarrow d\vec{s}=\frac{d\vec{r}}{V}$?

I am following a course which contains a part in statistical thermodynamics. One of the questions involves the partition function $Q_N$. I could not figure out the answer of the question myself, so I ...
9
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1answer
74 views

Evaluating by real methods $\int_0^{\pi/2} \frac{x^5}{2-\cos^2(x)}\ dx$

I'm sure you guys can briefly get the result by some methods of complex analysis, but now I'm only interested in real analysis methods of proving the result. What would you propose for that? ...
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0answers
17 views

Asymptotic expansion at infinity of integral function

Given $q\in(0,1)$ find $z$ such that $$ F(z)\equiv\int_{-\infty}^{z}\frac{e^{-\frac{y^2}{2 \sigma _{22}^2}} \text{erfc}\left(\frac{\rho \sigma _{11} y-\sigma _{22} V}{\sqrt{2-2 \rho ^2} \sigma ...
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2answers
35 views

meaning of integration

I read that integration is the opposite of differentiation AND at the same time is a summation process to find the area under a curve. But I can't understand how these things combine together and ...
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15 views

surface and cone integrals [on hold]

can someone take me through these two questions, I have the answers but not the steps and I have no idea how to even get started, thanks!
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1answer
34 views

Integrating $\int \sec^2(x) \tan(x) dx$ by trig substitution

I know I am supposed to integrate $$\int \sec^2(x) \tan(x) dx$$ by substituting $u = \tan(x)$ and get $du = \sec^2(x)$. However, why can't I use $u = \sec(x)$, $du = \tan(x) \sec(x)$?
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1answer
12 views

Poisson integral with discontinuous $U$

Let $U$ be a piecewise continuous function and bounded for all real numbers. Then define the Poisson Integral for the UHP to be (It can be deduce from the one for the unit circle). ...
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2answers
22 views

Proving complex integral on jordan region boundary equals to zero

Let $D\subset\mathbb{C}$ be a region bounded by jordan curve $\gamma$. Prove that: a. $\int_\gamma z \, dz=0$ b. $\int_\gamma \bar{z} \, dz\neq0$ (hint:$\bar{z}\,dz=(x-iy)(dx+i\,dy)$) ...
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1answer
28 views

U-Substitution. Why do you multiply the integrand by -1 in this case?

$$\int_0^{\pi/2} \! \frac{\sin x\cos x}{(4-\sin^2 x)^2} \:\text{d}x$$ set $u = 4-\sin^2 x$, therefore $du = -2 \sin x \cos x \text{d}x $ $$-\frac{1}{2} \int u^{-1/2} \text{d}u $$ Change the range ...
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1answer
49 views

Tough definite integration

For a curve given by: $x=e^{-t}\cos{2t}$, $y=\sin t$ R is the region bounding this curve, the x axis and the y axis (y-intercept is point a and x-intercept is point b). Find the exact coordinates ...
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2answers
30 views

How integrate $ \iint_{D} (\frac{x^2}{x^2+y^2})dA, \ \ \ \ D: x^2+y^2=a^2 \ \ and \ \ x^2+y^2=b^2, \ \ 0<a<b $

I'm trying to resolve this integral $$ \iint_{D} (\frac{x^2}{x^2+y^2})dA, \ \ \ \ D: x^2+y^2=a^2 \ \ and \ \ x^2+y^2=b^2, \ \ 0<a<b $$ I tried with polar coordinates: $$ x = r\cos{\theta} \\ ...
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2answers
16 views

solve polar coordinate integral

Evaluate $$\int_0^R\int_0^\sqrt{R^2-x^2} e^{-(x^2+y^2)} \,dy\,dx$$ using polar coordinates. My answer is $-\frac{1}{2}R(e^{-R^2+x^2}-1)$ but I want to confirm if that's correct And also, when I ...
3
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0answers
54 views

Solving double integrals numerically?

I have written this in way to make it as much as possible non-confusing. I will start describing my problem and I will walk you through my question, I have a double integration which I am trying to ...
2
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1answer
25 views

Indefinite integral and a trigonometric substitution

I have this integral: $\displaystyle\int\dfrac{x^3}{({\sqrt[2]{4x^2 + 9})^3}}\,dx$. I tried to solve it with a trigonometric substitituon but I can't get any result. I would appreciate if somebody can ...
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1answer
17 views

Arc length of this function

It is given that $x^2=(2y)^2$, he asks to give the arc length of this function, $1\leq x \leq 2\sqrt2$. Answer is $1/27 (19^{3/2} - 10^{3/2})$.
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0answers
14 views

Norm of arbitrary constant

I'm sitting in front of an exercise (basics in quantum mechanics), which wants me to check if the integral of a given function can be normed. One of those functions is the integral of zero from ...
4
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1answer
48 views

Calculating indefinite integral?

I want to calculate $$I_n = \int \frac{d\theta}{\sin^n(c\theta)\cdot \cos(c\theta)}. $$ The answer is $$-\frac{1}{c(n-1)\sin^{n-1}(c\theta)}+ ...
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0answers
23 views

Surface integral defined by a closed curve

So I know how to integrate over a surface defined by a parametric equation $$ \textbf{r}(u, v) = x(u, v) \textbf{i} + y(u, v)\textbf{j} $$ But what if the surface is defined as the area inside a ...
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0answers
29 views

integral sulotion over a and t [on hold]

what is the solution of this integral:$$\int^1_0 \frac{-2(t+a)+(1-a)}{((t+a)^2+(1-a)^2)^2} dt$$ canyou help me? that is a part solultion of a question which I should to solve it!
4
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0answers
84 views

How to compute or simplify this nasty integration?

Any hints on solving an integration of the following form, $$\int_{x}^{+\infty}\left(1-\frac{1}{1+sy^{-1}}\right) \left(\text{exp}(-\sqrt{y})+ y^{-\frac{1}{2}}(1-\text{exp}(-\sqrt[4]y)\right)dy $$ ...
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0answers
21 views

Gauss-Green Theorem from generalized Stoke's Theorem.

I am trying to deduce the next identity (Green-Gauss theorem) $$\int_\Omega \dfrac{\partial u}{\partial x_i} dx = \int_{\partial \Omega} uv_i dS$$ from the generalized Stoke's theorem for manifolds. ...
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0answers
8 views

lim Ln delta and d what do these mean? [on hold]

I need to know what does lim and Ln mean? And the difference between delta x and dX .I need these stuff in physical chemistry.
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2answers
27 views

Is the following true regarding integration?

Is the following correct? $$ \int_{x}^{+\infty} \left(f(u)-g(u)\right) du = \int_{x}^{+\infty} f(u)du - \int_{x}^{+\infty} g(u) du$$
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1answer
40 views

Prove the volume of a ball with radius approaching 0

Let f be continuous and let Br be the ball of radius r > 0 centered at $(x_0, y_0, z_0)$. Let V (Br) be the volume of Br. Prove that $$\lim_ {r\to0} \frac{1}{V(Br)}\ \iiint_{Br} \ f(x,y,z) dV = f(x_0, ...
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1answer
25 views

asymptotic expansion of this integral

How to get the asymptotic expansion for this integral $\int_{0}^{1}\exp(-x/t)dt $ in the limit $x\rightarrow 0$ ? I took $x/t=u$ and did integration by parts (IP) but if I keep doing IP, I get a ...
1
vote
1answer
33 views

Changing to spherical coordinates to evaluate the integral

$$\iiint_D \,dz\,dy\,dx$$ where the region $D$ is defined as followed: $$0<z<\sqrt{9-x^2-y^2}$$ $$0<y<\sqrt{9-x^2}$$ $$0<x<3$$ I got the corresponding spherical coordinates for ...
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0answers
44 views

Sorting out some integrals from physics

I'm doing some physics for a change, and I'm trying to sort things out a bit. From the definitions of mass, torque, momentum and angular momentum I've come up with the following integrals: ...
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0answers
17 views

Replacement in integral

Let's have integral $$ \int \limits_{-\infty}^{\infty}d^{4}k f\left(k^{2}, (k \cdot p )\right)k_{\mu} k_{\nu}, \quad d^{4}k = dk_{1}dk_{2}dk_{3}dk_{4}. $$ Here $f$ is integrable function, $(k \cdot p ...
0
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1answer
31 views

Proper definite of riemann integral (limit version)

I am sort of confused. Suppose we are given the series, $\displaystyle \lim_{n\to\infty}\sum_{k=1}^{n} \frac{k^{99}}{n^{100}}$ How can this be written as an integral, and what would the variable ...
3
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2answers
68 views

Solve $ \int_{0}^{\infty} \sin^2 \left(\frac{1}{x}\right)\mathrm{d}x$

I think this integral does not converge. I want to estimate downward the integral, but don't know how to.
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1answer
49 views

using complex or real analysis solve $\int_{0}^{\pi/2}\frac{x^m}{\sin x}dx$

closed form for $$\int_{0}^{\frac{\pi}{2}}\frac{x^m}{\sin x}\ dx$$ I slove it for some m but in general i failed. I tried by part , by substitution,by using $\sin x =\frac{e^{ix}-e^{-ix}}{2i}$ . I ...
2
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1answer
23 views

Integration of $x^a$ and Summation of first $n$ $a$th powers

I'm learning some discrete mathematics. I already knew a little (very little) calculus, and I noticed something. I think it's just a coincidence, so I'm sorry if this is a bad question. There are some ...
2
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2answers
53 views

Finding the $nth$ partial sum for $e^{-n}$

Here is the question: $$\displaystyle \sum_{n=1}^{\infty} e^{-n}$$ Instead of using the formula of $\large\frac{1}{1-r}$ I want to try to get the partial sums. $S_1 = e^{-1}$ $S_2 = e^{-1} + ...
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0answers
19 views

How to show that integration contours are related?

I have one geometry below in which the integration contours are shown with red and blue line. How I can show that the contour in blue line i.e (B to C) is with in the integration contour in red ...
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1answer
25 views

How to find a bound for these (simple) integrals

With help of $\int_{0}^{\infty} e^{-x^2} dx = \frac{\sqrt{\pi}}{2}$ and $\int_{0}^{\infty} e^{-x} dx =1$, I would like to know how to derive the following bounds: $$\int_{0}^m 4e^{-\frac{t^2}{8m}}dt ...
0
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1answer
45 views

If $f$ is continuous, then $\lim\limits_{n \rightarrow \infty} \int^b_a n(f(x+ 1/n)-f(x)) \lambda(dx) = f(b)-f(a)$

Consider a continuous function $f: \mathbb R \rightarrow \mathbb R$ and define $f_n: \mathbb R \rightarrow \mathbb R$ by $f_n(x) = n(f(x+1/n)-f(x))$. I want to show that for $a < b \in \mathbb R$ ...
20
votes
2answers
172 views

Prove the integral evaluates to $\frac{K}{\pi}$

Yesterday I received the following integral that might require some tedious steps to do $$\int_0^{\infty}{\small\left[ \frac{x}{\log^2\left(e^{\large x^2}-1\right)}- \frac{x}{\sqrt{e^{\large ...
0
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1answer
21 views

Why are trigonometric substitutions valid?

Within an integral, when you make a trigonometric substitution like $x = \sin(\theta)$ for $x$, aren't you changing the possible range of values for $x$? Aren't you limiting the possible values of $x$ ...
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1answer
35 views

Is there a function $f \gt 0$ such that $\int f dx=0$?

If $$f \ge 0$$, then $$\int f dx \ge 0$$. There is a function such that $$f \gt 0$$ but $$\int f dx=0$$?
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votes
2answers
25 views

How to evaluate this path integral?

So I know that the integral is $$\int_1^e f(x(t), y(t), z(t))(||c'(t)||) \:dt$$ I set this to$$\int_1^e\frac{1}{t^3}\sqrt{\frac{1}{(ln10*t)^2}+1}\; dt$$ I found this too hard to integrate by hand, ...
0
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1answer
30 views

Integral of the log is less than the integral of the log of the average value

This is an interesting property that I came across while reading an old proof on this website. The poster didn't really explain it, so I thought I might ask. We suppose $u$ is a positive measure on ...
1
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0answers
11 views

Poisson Integral, when $U$ is discontinuous

So I am working on the following problem. Let $U$ be a piecewise continuous function and bounded for all real numbers. Then define the Poisson Integral for the UHP to be (It can be deduce from the one ...
0
votes
1answer
21 views

Find the mass of the disk. - Double Integration Problem - Calculus 3

A disk of radius 5 cm has density 10 g/cm2 at its center, density 0 at its edge, and its density is a linear function of the distance from the center. Find the mass of the disk. my answer: 157.08g ...