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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How Prove this integral is diverge $\int_{0}^{1}\dfrac{dx}{\ln{x}\ln{(1-x)}}$

Show that this following integral is divergent (or diverges, if you prefer) $$\int_{0}^{1}\dfrac{dx}{\ln{x}\ln{(1-x)}}$$ I know when $x=0,1$ are singularities of the function and I want use this ...
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1answer
27 views

Question about substitution method in integration

It is common that we replace $\int u(x)v′(x)\mathrm{d}x$ by $\int u \mathrm{d} v$ where both $u$ and $v$ are continuous functions of $x$. My question is, must we ensure that $u$ can be written as a ...
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0answers
23 views

Solving an Integral by Summation [on hold]

My final answer for this question was 90 but I'm not quite sure if I'm even doing it right... I was wondering if anyone could help me solve this for me to check my work against. ...
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1answer
28 views

Integrand for a set of points

I need help finding what I should be integrating when the question asks to find the double integral to find the volume of the tetrahedron given the points $(0,0,0),(3,0,0),(2,1,0),(3,0,4)$. Would the ...
3
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2answers
51 views

Using an Integral to Solve for a Variable a

I am struggling to use the following equation: $$ \int_0^a \sqrt{a^2-x^2}\,\,\text{sgn}(|x|-1)\, dx = 0 $$ where $a > 1$, to deduce that $a = \text{cosec}(\frac{\pi}{4} - \frac{\alpha}{2})$, ...
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1answer
25 views

Function of a surface area?

$$\iint\limits_s {y \cdot dS}$$ $$z=x+y^2$$ $0 \le x \le 1$ and $0 \le y \le 2$ if you graph $z$, then you get a surface in a 3d scalar field from the ranges above. How does just $y$ in the ...
3
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0answers
34 views

Fubini's theorem application proof check

I have proven a problem but I am unsure whether it is correct because the proof seems so simple that I think I might be mistaken. Please be kind to comment on my proof and tell me whats wrong with it. ...
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2answers
44 views

How do you apply $u$-substitution to the integral $\int v\sqrt{2v^2+1}\,dv$?

When doing $u$-substitution of the following integral problem, does the $v$ disappear because it has a value of $1$? $$\int v\sqrt{2v^2+1}\,dv$$
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1answer
53 views

Taylor series of a definite integral

Consider the function of a parameter $\alpha > 0$, given by $$f(\alpha) = \frac{2}{\sqrt 2\pi} \int_0^\infty e^{\dfrac{-x^2}{2\alpha^2}}\cosh(x)\log\cosh(x) dx.$$ From Wolfram-alpha, it seems that ...
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0answers
13 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
71 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 ...
2
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5answers
56 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
13 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) = ...
2
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1answer
44 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, ...
1
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2answers
60 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
26 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 ...
1
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1answer
44 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
19 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
67 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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0answers
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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
40 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
58 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
37 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
50 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 ...
0
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0answers
20 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
52 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
64 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
votes
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
59 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
votes
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
71 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
votes
2answers
79 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
votes
4answers
51 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
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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
32 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
47 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
16 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 ...
0
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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 ...
0
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0answers
45 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 ...
2
votes
2answers
86 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
27 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
votes
3answers
57 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
29 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
votes
3answers
59 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 ...