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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1
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2answers
13 views

Indefinite integrals with rati0nal and polynomial functions and Substituion

I am totally confused with the substitution method of evaluating indefinite integrals, especially those with rational functions and polynomials. I have 2 cases, which if I made to understand, would ...
0
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2answers
51 views

How do I solve $\int_{0}^{\infty} \frac{\ln(x)}{1+x^{2}}\,dx$?

If we first split the integral into two: $$\int_{1}^{\infty} \frac{\ln(x)}{1+x^{2}}\,dx$$ and $$\int_{0}^{1} \frac{\ln(x)}{1+x^{2}}\,dx$$ Let $x = 1/u$ and $dx = -1/u^2 du$, then we have: ...
1
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1answer
25 views

Deriving a joint cdf from a joint pdf

I see that a similar question was asked last year, but I am still confused. I have $f(x,y) = 2e^{-x-y}$, $ 0 < x < y < \infty $ and need to find the joint CDF. I have a solution that ...
0
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1answer
16 views

is it possible to intergrate this function to get x(t) and y(t)?

say you have a function as below; $d^2V(t)/dt = -B^2V(t)$ B is a constant Initial conditions $V_x(0) = V$, $V_y(0) = 0$ I can't see how to integrate to get x(t) and y(t); I ended up with ...
0
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1answer
21 views

Calculating surface area

I have the following surface in $$R^3:{(x,y,z),(x^2 + y^2 + z^2)^2 = a^2(x^2 - y^2) \ ,\ x,y >=0}.$$ I want to find it's surface area. I've tried using spherical coordinates but calculating the ...
0
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0answers
17 views

Does isometry preserve volume on open sets?

Suppose there are two open sets $A,B$. $h$ is an isometry. And the function $h$ maps $A$ to $B$; $h(A)=B$. I need to show that isometry is volume preserving. My idea is that if the continuous ...
7
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0answers
42 views

how to compute this limit

compute $I=\lim\limits_{n\to+\infty}\sqrt[n]{\int\limits_0^1x^{n+1}(1-x)\cdots(1-x^n)dx}$ attempt: i tried to evaluate the integral $$\begin{align} ...
0
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1answer
22 views

integration and convolution

Please can some one help me on the following integration. $$ G(\nu)=\frac{1}{\Delta t}\int_{t_a - \frac{\Delta t}{2}}^{t_a + \frac{\Delta t}{2}} f(t_a -t)e^{-2\pi\nu it}dt $$ where ...
1
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2answers
53 views

Solving $\int^{\pi}_{ 0.5\pi} \frac{dx}{1-\cos x}$

Evaluate $$\int^{\pi}_{ 0.5\pi} \frac{dx}{1-\cos x}$$ This is my attempt: $$\int^{\pi}_{ 0.5\pi} \frac{dx}{1-\cos x} = ...
0
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2answers
27 views

derivative and integral as opposite operations

Consider: $$\lim_{y\to\infty} \left( \int_0^y f(t)dt \right)' = \lim_{y\to\infty} f(y)$$ So the integral and the derivative cancel each other, but why is it happened to be that it equals to the ...
1
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3answers
55 views

Evaluation of integral $\int_{-\infty}^{+\infty} xe^{-|x|}\,dx$ is not $0$

Given $$f(x)=\frac12e^{-|x|}, -\infty \le x \le +\infty$$ $$\int_{-\infty}^{+\infty} x f(x)\, dx= -\frac12\int_{-\infty}^{+\infty} x (-e^{-|x|})' dx=-\frac12\bigg(-xe^{-|x|} + ...
0
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0answers
23 views

indefinite integral problem: help needed

What will be the integral with respect to $t$ of: $$\frac{dA}{dt} = cx(t)y(t),$$ where $c$ is a constant and $x$ and $y$ are functions of time ($t$). Is there any other method besides inegration by ...
0
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0answers
15 views

contour integral and limit: What is the condition of the interchange the order?

In real real analysis sense, the interchange between limit and integral is hold when integrand is uniformly converges. $i.e$ \begin{align} \int \lim f = \lim \int f \end{align} Here i want to ...
0
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3answers
35 views

Divergence/convergence of an integral

I am told that the following integral converges for $1<n<3$. $$ \int_{-\infty}^{+\infty} (1-e^{ix}) |x|^{-n} dx $$ I am a bit baffled. Anyone with a clue or where to start with this in order to ...
5
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4answers
49 views

Prove that $\int_0^1 \frac{\ln x}{x-1} dx$ converge.

Prove that $\int_0^1 \frac{\ln x}{x-1} dx$ converges. We cannot apply Abel's/Dirichliet's tests here (For example, Dirichliet's test demands that for $g(x)=\ln x$, $\int_0^1 g(x)dx < \infty$ ...
4
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5answers
87 views

Solve $\int_0^\infty \frac{\ln x}{x^2+4} \,\mathrm{d}x$

(Stanford Math Tournament 2012 #8) I tried rewriting the denominator as $4\left(\frac{x}{2}^2 + 1\right)$ and then integrating by parts, but that got me nowhere... I then tried the substitution $x = ...
-1
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0answers
24 views

analytic solution of a definite integral

Integrate the following $$\int_0^\infty \alpha\,\beta\, c\, k\, x^s\, x^{c-1} (1+x^c)^{k-1} \left[(1+x^c)^k-1\right]^{-\beta-1} \left[1+\gamma ...
2
votes
2answers
380 views

Splitting an integral

Why is the following equality true? $$ \int_1^{2e} \left| \ln x - 1 \right| dx = \int_1^e(1-\ln x) dx + \int_e^{2e} (\ln x - 1) dx$$
1
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1answer
35 views

Show how $\frac{\partial}{\partial x} \left[\int_0^x (x-t)g(t)\,\mathrm{d}t\right] = \int_0^x g(t)\,\mathrm{d}t$

It has something to do with the second part of the Fundamental Theorem of Calculus right? I've always had trouble with this theorem ever since I learned it several years ago :\ Would somebody please ...
0
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1answer
34 views

Showing $\sum_{n\in\mathbb{N}}\int{|f_{n}-f|d\mu}<\infty$ implies $f_{n}\rightarrow f$ almost everywhere.

Let $(f_{n})_{n\in\mathbb{N}}$ be a sequence of integrable functions and $f$ an integrable function. I have to show that, if $$ \sum_{n\in\mathbb{N}}\int{|f_{n}-f|d\mu}<\infty, $$ then ...
1
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1answer
17 views

Integration with respect to a measure

I am trying to get an explanation in words, or math, of what the $d\mu$ means in an integration statement. Such as: $$\int f \ d\mu$$ How does the measure change our old "calculus" notion of ...
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0answers
20 views

This is a 2+1 D problem. I know a little about contour integration. Please suggest how may I proceed. [on hold]

Kindly check it.I don't know what to do with this delta and etc etc $\int \frac {e^{-i\vec{k}.( \vec{x}'-\vec{x})}e^{-ik^{\circ}x_{\circ}}\delta(k^{\circ})}{k^2-\mu^2}d^2kdk^{\circ}$
1
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1answer
25 views

Proving an identity using Riemann-Stieltjes Integration?

Prove the following identity using Riemann-Stieltjes Integration: $$\sum_{n=1}^N \frac{1}{n^s} =\frac{1}{N^{s-1}} + s \int_1^N \frac{\lfloor x\rfloor}{x^{s+1}}dx$$ Here's what I have so far: $$ ...
5
votes
2answers
26 views

Conceptual question on substitution in integration [duplicate]

In calculus we learn about the substitution method of integrals, but I haven't been able to prove that it works. I mainly don't see how manipulations of differentials is justified, i.e how $dy/dx = ...
0
votes
1answer
53 views

Integral $\int_0^\infty e^{-x/2}x\log(1+kx^2)\,dx$

How to evaluate: $$\int_0^\infty e^{-x/2}x\log(1+kx^2)\,dx$$ Basically am evaluating value of $\log(1+c\chi^2)$ where $\chi^2$ is $\chi$-squared distributed
2
votes
1answer
39 views

Integral $\int\frac{(\ln x)^{10}}{x}\,dx$

$$\int\frac{(\ln x)^{10}}{x}\,dx$$ All I know is that I am supposed to substitute $u=\ln x$. But can someone please explain to me how to find the anti derivative of $(\ln x)^{10}$. I think we are ...
0
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3answers
72 views

How $\int_{-\infty}^{\infty}\frac{dx}{1+x^2}$ exists?

How $$\int_{-\infty}^{\infty}\frac{dx}{1+x^2}$$ exists? It is difficult question to me. i have tried to evaluate by using fact that $$\int_{-\infty}^{\infty} f(x) \ dx =\int_{-\infty}^{0} f(x)\, dx ...
2
votes
0answers
37 views

how to calculate this line integral $\int_{0}^{2\pi} (16\sin^2 3t +16\cos^2 4t)\sqrt{(144\cos^2 3t +256\sin^2 4t)}dt$

I am working on a line integral to calculate the amount of chocolate to cover a pretzel. the density of the pretzel is given by this formula $\lambda=3(x^2+y^2)$ and the parameter equation of a ...
1
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2answers
33 views

When does a double integral represent a surface area, and when does it represent a volume?

When does $\int_Af(x,y)dA$ represent a surface area geometrically, and when does it represent a volume? In my lecture notes I'm told it represent the volume underneath the surface $z=f(x,y)$, but I've ...
1
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0answers
30 views

line integrals explanation

I am very new to this so sorry if it is obvious. Compute the line integral $\int Fdr $ where $F(x,y)=(x^2y,y^2x)$; $r(t)=(\cos t,\sin t)$; $t\in[0,2\pi]$. So what I would do is find $r'(t)=(-\sin ...
2
votes
1answer
78 views

Derivative under integral mixed with…

$$f(x,y)=\int_{e^{4y}}^{\ln^3(x)}{\frac{\sin(t)}{t}\,dt}$$ Whats the derivative $\frac{d f}{d t}$, if: $$x(t)=\cos(2+6t).4t^2$$ $$y(t)=\ln(2r+7e^{5t})$$ Really not much to say about this problem ...
1
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1answer
32 views

Is this theorem about integration with substitution wrong?

A theorem in my book states: If $g$ is differentiable, f is continuous, and F is an antiderivative of f, then : $\int f[g(x)]g'(x)dx=F[g(x)]+C$ The reason I am asking if this is correct, ...
1
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1answer
36 views

When does the integral converges?

For what $\alpha, \beta$ the integral $$\int_0^\frac{\pi}{2} \frac{(\frac{\pi}{2} - x)^\alpha}{(\cos x)^\beta} dx$$ converges? So first I've approved (using WolframAlpha) that $\frac{\pi}{2} - x ...
2
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1answer
24 views

Stuck on an integration question…

$$\int x^{-\frac{1}{2}}\cosh^{-1}(\frac{x}{2}+1)dx$$ The answer I should get is $$2x^{\frac{1}{2}}\cosh^{-1}(\frac{x}{2}+1)-4(x+4)^{\frac{1}{2}}$$ but I keep going wrong. Can someone show me how to ...
1
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2answers
58 views

How can I prove this integral?

I have to use the identity $b^4-a^4=(b-a)(b^3+b^2a+ba^2+a^3)$ to prove that: $\int_b^ax^3dx=\frac{b^4-a^4}{4}$. I know that you can just do $F(b)-F(a)$ and since the integral of $x^3$ is ...
0
votes
2answers
34 views

calculate $\int_{0}^{2\pi}\frac{1-\sin(t)}{2-\cos(t)}dt$

I need to calculate $\int_{\gamma} \frac{1-\sin(z)}{2-\cos (z)}dz$ where $\gamma$ is the upper hemisphere of the circle with center $\pi$ and radius $\pi$, with a positive direction. The original ...
2
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2answers
26 views

Improper integral calculation - limit at infinity

Will you please help me prove the following limit is zero ? $$\lim_{x \to \infty} \int_0^{\infty} \frac{1-e^{-u^4}}{u^2} \cos(x u) du. $$ Thanks in advance
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2answers
33 views

Evaluating $\int \frac{(1+\cot^2 x)(\cot x)}{\csc x}dx$ [on hold]

Please could someone help with $$\int \frac{(1+\cot^2 x)(\cot x)}{\csc x}dx$$? Thanks
0
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3answers
35 views

Integral $\int \frac{1}{(4x^2-8x+3)^{1/2}}$ [on hold]

Please could someone help with the integration of $\frac{1}{(4x^2-8x+3)^{1/2}}$? Thank you.
1
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1answer
15 views

Prove that for every $T > \frac{\pi}{2} $, $\int_{\frac{\pi}{2}}^T \frac{cos(x)}{x}dx < 0$

I tried doing integration by parts a few times, after doing it 3 times I get the following expression: $$ \int_{\frac{\pi}{2}}^T \frac{cos(x)}{x}dx = \frac{sin(T)}{T} - \frac{1}{\frac{\pi}{2}} - ...
3
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1answer
40 views

Average distance to a random point in a rectangle from an arbitrary point

I'm interested in the mean distance between an arbitrary 2D point, $(p, q)$, and a uniformly distributed point inside a rectangle defined by the lower left and upper right vertices $(x_0, y_0)$ and ...
1
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2answers
55 views

How to solve $\int{\frac{1}{\sqrt{3-2x-x^2}}\,dx}$?

$$\int{\frac{1}{\sqrt{3-2x-x^2}}\,dx}$$ I tried to do it by substitution with no sucess. Anyone can solve it?
2
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4answers
140 views

Anyone can integrate $e^{-\frac{x^2}{3}}$ by hands?

I just used wolfram integral calculator and the result is weird, there is something called error function. $$ \int_{-\infty}^\infty e^{-\frac{x^2}{3}}\,\mathrm dx $$ Hint says that change of variable ...
0
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3answers
31 views

Line integral of two intersecting spheres

How can I find the length of the line formed by two intersecting unit spheres shifted a distance x from each other? Any suggestions to approaching the problem is also greatly appreciate! Thanks!
4
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2answers
27 views

Investigate the convergence of $\int_1^\infty \frac{\cos x \ln x}{x\sqrt{x^2-1}}$

Investigate the convergence of $$\int_1^\infty \frac{\cos x \ln x}{x\sqrt{x^2-1}}$$ so first of all let's split the integral to: $$I_1 = \int_1^2 \frac{\cos x \ln x}{x\sqrt{x^2-1}}, I_2 = ...
2
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0answers
22 views

Check my answer - complex analysis, using residue and rouche's theorem

I was asked the following questions and I am unsure of my solutions, any advice would be appreciated, maybe there is a better way of doing this. Question: We are given $f(z)=2z-\sinh (z)$ defined on ...
2
votes
3answers
63 views

Limit evaluation with integral

Evaluate the limit $$\lim_{n\to\infty} \int_0^1 n^2x(1-x^2)^n dx$$ My Proof: We may look at $n$ as a constant and evaluate the integral $\int_0^1 x(1-x^2)^ndx$ (I already moved out the $n^2$). ...
3
votes
2answers
43 views

Antiderivative of $\frac{\sqrt{4-x}}{x\sqrt{x}}$

I need help to find the antiderivative of the function $\displaystyle x \, \mapsto \, \frac{\sqrt{4-x}}{x\sqrt{x}}$ on $]0,4[$. I have tried the change of variables $u = \sqrt{4-x}$ but it didn't ...
1
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1answer
26 views

Compute integral given 2 other integrals

I want to know which solution is correct. The question states: If f is an integrable function on [1,3], and if $$\int_1^2f(x)dx=4 \space\space\space\space\space\space and \space\space\space\space ...
0
votes
1answer
33 views

What is wrong with my integral solving

Consider the integral $$\int{ \frac{x^2+2x+8}{(x^2-2x)(x^2+4)}}dx$$ I simplify it to $$\int\frac{x^2+2x+8}{x(x-2)(x^2+4)}$$ Then I try to solve it in sum of partial fraction which gives me ...