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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6 views

Show that the function $f(x,y)=\int_b^yf_2(a,t)\ dt + \int_a^xf_1(t,y)\ dt $ is a potential function

Let $F=(f_1,f_2)$ be conservative over the open rectangle: $$R=\{(x,y):|x-a|<r,|y-b|<r\} $$ I need to show that the function $f(x,y)=\int_b^yf_2(a,t)\ dt + \int_a^xf_1(t,y)\ dt $ is a ...
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3answers
59 views

How to integrate $1/(u^2 + u^4)$ du?

I did a trig substitution with $x = \tan \theta$ followed by a regular $u$ substitution and I got the integral down to $$\int \frac1{u^2 + u^4}\mathrm du$$I just need a reminder of what this would be ...
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5answers
52 views

How to integrate $\int_1^\infty \frac{dx}{x^2\sqrt{x^2-1}}$?

How to integrate $$\int_1^\infty \frac{dx}{x^2\sqrt{x^2-1}}$$ I tried both $t=\sqrt{x^2-1}$ and $t=\sin x$ but didn't reach the right result.
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0answers
12 views

Minimization of the integral with respect to a parameter

Intro Let $f$ be a a real-valued function parametrized by a parameter $\alpha \in \mathbb{R}$ and let $J\colon \mathbb{R} \to \mathbb{R}$ be a functional defined as follows: $$J(\alpha) = ...
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1answer
37 views

Prove or disprove following integral.

Assume $L$ a constant, and assume $x$ real. Is the following equation true? $$ \int_{-\infty}^\infty\frac{1}{k^2}\exp(-ikx)dk = \frac{L}{|x|} $$ If it is true, find the value of $L$. If it is not, ...
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2answers
51 views

Find $\int_0^2 \arctan(\pi x)-\arctan(x)\, \mathrm dx$

Find $$\int_0^2 \arctan(\pi x)-\arctan(x)\, \mathrm dx$$ The hint is also given : Re-write the Integrand as an Integral I think we have to Re-write this single integral as a double integral and ...
2
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1answer
22 views

Complex integral and Laurent series

Could you help with solving this complex integral: $$\int_C z^3\exp{\left(\dfrac{-1}{z^2}\right)} dz$$ where $C$ is $|z|=5$. I am expecting that the Residue Theorem will be needed. The answer should ...
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1answer
43 views

Integration (Cosine Function)

Ive been doing some integration study and ive been caught by this question. Anyone have any ideas? Thanks. Apologies on how the question is presented, im no quite sure how to do it properly yet. - ...
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0answers
16 views

Banach Spaces: Improper Riemann Integral

Disclaimer This thread is related to: Stone's Theorem Definition Given a measure space $\Omega$ and a Banach space $E$. Consider functions $F:\Omega\to E$. Denote the measurable subsets of finite ...
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3answers
61 views

Example of Riemann integrable $f: [0,1] \to \mathbb R $ whose set of discontinuity points is an uncountable and dense set in $[0,1]$ [on hold]

Give example of a function $f: [0,1] \to \mathbb R $ which is integrable ( Lebesgue or Riemann , if possible , both) but whose set of discontinuity points is an uncountable set and dense in $[0,1]$ ...
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22 views

Solving second order ordinary differential equations [on hold]

I would like you to ask you to solve the following two ordinary differential equations. It is highly appreciated your kind consideration in advance. 1.Solve y"(x)=f(x) 0<=x<=1, where f is ...
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1answer
20 views

prove integration formula relating to derivatives

Could any one help me solve this problem ? it is from Apostol's calculus volume 1
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0answers
33 views

Hard sum with harmonics numbers

Prove or disprove that $S=\displaystyle\sum_{n=1}^{\infty}\frac{{H_n^{2}}~{H_n^{(2)}}+3{H_n^{(4)}}}{n~2^n}=\frac{25}{16}\zeta(5)+\frac{7}{8}\zeta(2)\zeta(3)$.
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1answer
54 views

$ 0 \le f(x) \le 1 $ for $ 0 \lt x < 1 \implies \int_0^x f(t)t ~dt \le x^2 $ for all $ x\in(0,1) $?

I have the following implication, and I need to determine whether it's true: $ 0 \le f(x) \le 1 $ for $ 0 \lt x < 1 \implies \int_0^x f(t)t ~dt \le x^2 $ for all $ x\in(0,1) $ I tried solving ...
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0answers
35 views

$F(x) = \int_0^x f(t)~dt \implies F(1)=f(0)+\int_0^1(1-t)f'(t)~dt$?

f is differentiable and has a continuous derviative, and $F(x) = \int_0^x f(t)~dt$. Based on this assumption, I have the following statement which I need to determine whether it's true or false: ...
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2answers
46 views

f is even or odd, prove that f^2 is even

I need to verify whether a statement is correct or false. The statement is as following: If the function f is either odd or even, then the function f^2 is even. To my understanding, the statement is ...
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0answers
19 views

big $\mathcal O$ for number of prime in an interval?

According to von Koch 1991, if the Riemann hypothesis is true, then the for the prime counting function $$\pi(x)=Li(x)+\mathcal O(\sqrt x \log x)$$ I am trying to understand how to deal with the ...
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0answers
18 views

Integrals: Average(f)*Average(g)=Average(f*g) [on hold]

So I've got everything but question #3 here. I understand that it isn't simply (1/4)(1/4)=16. And also not (1/4)(1/4)(1/4)=1/64. But I can't think of what else it might be. It isn't discussed in the ...
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2answers
50 views

Evaluate $\displaystyle\int_{-\infty}^{\infty}\frac{dx}{(1+x^2)^2}$ using residue method [on hold]

This is a real integral but I want to evaluate it using residue integration method $$\int_{-\infty}^{\infty}\frac{dx}{(1+x^2)^2}$$
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1answer
47 views

Integration by Substitution, can't solve (Working Added )

My Working: $$\displaystyle dx = du/2x$$ sub dx and U into equation $$ x^2 \int x(U)^{3/2} du/2x$$ Eliminate x $$ x^2/2 \int (U)^{3/2} du$$ $$ x^2/2. [2(U)^{5/2}/5]$$ then $$ ...
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4answers
63 views

Evaluating $\displaystyle \int\frac{1}{\sqrt{(x-2)(5-x)}}\,dx$ using trigonometric substitution [on hold]

Using Substitution Integral Method, compute $$\displaystyle \int\frac{1}{\sqrt{(x-2)(5-x)}}\,dx$$ (let $x=2\cos^2\theta+5\sin^2\theta$)
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0answers
9 views

Recursive formula for Laguerre guassian integral?

The integral of interest is: $ I_{l, m} = \int_{u0}^{u1} u^{(l+1)/2} e^{-u/2} L_m^l(u) du $ where $L_m^l$ is the laguerre polynomial. What I'm interested in is getting some relation to lower order ...
3
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3answers
227 views

Indefinite integral of a simple function

$\int 2(1 + \tan^2 x)$ My work : $2(1 + \tan^2(x) = 2 + 2\tan^2x$ $2x + \frac{2}{3}$ $\tan^3(x) \cdot \ln|sec(x)| + C$ The answer says no, after multiple tries :(
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1answer
32 views

Expressing limit of sum definite integral

Evaluate limit by expressing it as a definite integral. ...
2
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4answers
57 views

Derivation of the integral

Evaluate $$\large\frac{d}{dx}\int_{0}^{\large\int_0^{e^x}{\cos (s)\,\mathrm ds}}\sec(t^2)\,\mathrm dt$$ I got the answer to be $$e^x\cdot\sec(\sin^2(e^x))\cdot \cos(e^x)$$ but do not know if ...
2
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1answer
13 views

Singular chain complex for integration - pinching on boundary

Singular chain complex, as far as topology are concerned, is just continuous map from standard simplex, and the choice of using simplex over other shape is immaterial. But for integration on manifold, ...
3
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1answer
68 views

If nonnegative $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$, then $\int_0^1 \Big| \frac{f''(x)}{f(x)} \Big| \,dx >4$

Assume that $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$ and $f$ is positive on the interval $(0,1)$ and $0$ at the endpoints. I want to prove that $$\int_0^1 \Big| \frac{f''(x)}{f(x)} ...
3
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2answers
78 views

$\frac{1}{x^2} \int xe^x dx$ without using integration by parts

On a test i just had, i needed to solve a differential equation which lead me to having to find the result of $$ \frac{1}{x^2}\int xe^x dx $$ I then attempted to do this integral without integration ...
2
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4answers
50 views

Integration by parts of $\cos(x)e^{-x}dx$

I do the integral but I end up getting the original $\cos(x)e^{-x}dx$ on both sides and canceling them out resulting in no solution. Can I get a step by step break down of how to solve?
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0answers
40 views

How is Riemann–Stieltjes Integration insufficient for developing modern probability theory?

If we consider Riemann–Stieltjes integration then it can perfectly account for mixed probability distribution (a continuous R.V with some point mass). So why would we still need Lebesgue Integration ...
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1answer
31 views

Problem with this question on solid of revolution

Calculate the volume of a revolution solid obtained by rotation around the x-axis, the region bounded by the graph of $y=e^x$, $-1\le x \le1$ and the x-axis. Thanks in advance, and sorry about my ...
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1answer
45 views

Recapitulated: Stone's Theorem Integral

This problem grew out from: Stone's Theorem Integral For a definition, a nonexample and a comparison see: Generalized Riemann Integral: Definition Generalized Riemann Integral: Nonexample ...
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0answers
15 views

Simple indefinite integral of a vector function

I am having trouble with this simple integration. I am not sure of the process or steps to follow to solve this type of problem: If $\mathbf{V}(t)$ is a vector function of $t$, find the indefinite ...
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0answers
20 views

When substituting in integration, do you have to change the limits of integration so long as you keep it consistent?

I have this integral: In order to solve for it, I have to substitute: t=tan(theta) dt=(sec(theta))^2 d(theta) When substituting that, I know I have to change the limits of integration within ...
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0answers
44 views

Piecewise vs Continuous Integration

I have the following data: Daily spend on marketing Daily gain of fans because of that spend on marketing ('billed' fans) The 'organic' daily number of fans for the same period above (ie free ...
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0answers
45 views

Integration of a polynomial

I am facing a problem in finding the integral $$\int\frac{r^2}{-C r^3 + r^2 -2 M r +Q^2}\,dr$$ Here M, Q, and C are parameteres (to be fixed later). Could anybody Please help me in finding it? I ...
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2answers
85 views

Calculation of $\int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x} dx\;,$

Calculation of $\displaystyle \int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x} dx\;,$ given that $ a>b>0$ $\bf{My\; Try::}$ Let $\displaystyle I = \int_{0}^{\pi}\frac{\sin^2 ...
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1answer
24 views

Uniform convergence and integrability

If $(f_n)_{n \in \Bbb N}$ converges to $f$ uniformly and each $f_n$ integrable would it imply $f$ is integrable and $$\lim_{n \to \infty}\int f_n = \int f$$ In case each $f_n$ is nonnegative ...
3
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3answers
56 views

Is $\int_1^\infty \frac{\log(x-1)}{x(x-1)}\,dx$ convergent?

Does the following integral $$\int_1^\infty \frac{\log(x-1)}{x(x-1)}\,dx$$ converge? If it is convergent can we compute it?
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1answer
28 views

Counting function for the number of zeros of a continuous positive function?

Let $f(x)$ within $x\in[a,b]$ an absolute continuous function with $f(x)\geq0$ $f(x_m)=0$ for all absolute minima $x_m$ no other zeros than at $x_m$ I am trying to define a counting function for ...
3
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3answers
58 views

Problems with this integral $ \int \sqrt{1 + {1 \over t^2} + {2 \over t}} dt$

$$ \int \sqrt{1 + {1 \over t^2} + {2 \over t}}\,\mathrm dt$$ I tried making substitution, using $ u=1 + \dfrac{1}{ t^2} + \dfrac{2 }{ t} $, then , $dt=\dfrac{du}{-2\left({1 \over t^3 }+ {1 \over ...
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2answers
104 views

Evaluating $\int^{4}_{1} \sqrt{1+\left(\frac{1}{2\sqrt{y}}-7\right)^2} dy$

I was trying to find arc-length of $x = \sqrt{y}-7y$ So basically right now I am stuck with this $$\int^{4}_{1} \sqrt{1+\left(\frac{1}{2\sqrt{y}}-7\right)^2} \,\mathrm dy$$ $$\int^{4}_{1} ...
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2answers
28 views

Arc length of a curve which already has an integral sign

This one here was tricky because the function already has an integral sign. My guess is that I need to evaluate the indegral where $x=4$ so that i get $y=f(t)$ and after that apply the Arc Length ...
6
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2answers
59 views

Integration $\frac{1}{2\pi}\int_{-\pi}^{\pi}(x-a)^ke^{-i\omega x}dx, \ \ \ \ a\in\mathbb R$.

Give a compact form for the solution of integral: $$\frac{1}{2\pi}\int_{-\pi}^{\pi}(x-a)^ke^{-i\omega x}dx, \ \ \ \ a\in\mathbb R,k\in\mathbb N$$ any suggestions please?
2
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1answer
24 views

Partial Derivative of a nonexistant variable?

I am wondering how I would find the partial derivative of $z = g(r, \theta) = \theta$ with respect to both $r$ and $\theta$. I realize that if you take the partial in respect to $\theta$, it is 1. I'm ...
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0answers
38 views

Antiderivative of $|x − 2| + |x − 3|$ [on hold]

Find the most general antiderivatives of the following function. $$|x − 2| + |x − 3|$$ I started with showing that the antiderivative for $|u|$ is $\dfrac{u|u|}2$. How to proceed then?
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0answers
26 views

The set composed of domain and codomain of integrable function measure zero

There is this problem which I have constructed a plan to prove, and I am stuck. If anyone could see my plan and tell what is wrong about it I would be very thankful. Let $f: Q \to [0,1]$ be ...
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1answer
51 views

Taking the Derivative of $F(x)=\int_0^x f(t)\,dt$ [on hold]

Let $F(x)=\int_0^x f(t)\,dt$ What is the derivative of $F(x)$? I desperately need guidance!
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1answer
34 views

Lebesgue integration in one variable

I have studying the conditions for the existence of the Lebesgue integral. Generally, to show that existence of the integral of a function on an unbounded interval, one can integrate and take ...
0
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1answer
31 views

Verify Green's Theorem for region bounded by the lines $x=2$, $y=0$, $y=2x$

Verify Green's Theorem for the region D bounded by the lines $x=2$, $y=0$, $y=2x$ and the functions $f(x,y)=(2x^2)y$, $g(x,y)=2x^3$. I have been trying this question for far too long and I can't ...