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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51 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: ...
1
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2answers
62 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 ...
1
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1answer
38 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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1answer
60 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
147 views

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

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
18 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
252 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 :(
0
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1answer
65 views

Expressing limit of sum definite integral

Evaluate limit by expressing it as a definite integral. ...
3
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4answers
94 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
37 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
95 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
105 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
60 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?
4
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0answers
73 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 ...
0
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1answer
40 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 ...
2
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2answers
83 views

Stone's Theorem Integral: Avanced Integral

Reference This problem grew out from: Stone's Theorem Integral: Basic Integral Problem Given the real line as measure space $\mathbb{R}$ and a Hilbert space $\mathcal{H}$. Consider a strongly ...
1
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0answers
23 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
61 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 ...
3
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2answers
129 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 ...
-1
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1answer
40 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 ...
4
votes
3answers
80 views

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

Does given integral $$\int_1^\infty \frac{\log(x-1)}{x(x-1)}\,dx$$ converge? If it is convergent can we evaluate it's value?
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1answer
36 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
84 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
117 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} ...
0
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2answers
33 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
votes
2answers
87 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
26 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 ...
2
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0answers
44 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 ...
0
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1answer
40 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
61 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 ...
2
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1answer
37 views

Inequality involving Holders Inequalities

Suppose $f\in L^p(\mathbb{R})\cap L^\infty(\mathbb{R})$ for some $p>2$, show that $||f||_{p}\leq ||f||_2^{2/p}||f||_{\infty}^{1-2/p}$ I tried to write $|f|^p=|f|^{\frac{p}{2}}|f|^{\frac{p}{2}}$ ...
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1answer
40 views

Prove that the antiderivative of an integrable function is both bounded and integrable

Let $f: [a,b] \to \mathbb{R}$ be a bounded function which is also integrable. Define $F: [a,b] \to \mathbb{R}$ by $$F(x)=\int_{a}^xf(t)\ dt$$ To prove that $F(x)$ is also bounded and integrable I ...
4
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7answers
234 views

Show that $\int_0^\infty \frac{\sin (\lambda x)}{e^x} \, \mathrm dx =\frac{\lambda}{1+{\lambda^2}}$

$$\int_0^\infty \frac{\sin (\lambda x)}{e^x} \, \mathrm dx =\frac{\lambda}{1+{\lambda^2}}$$ My intuition telling me there might be an $\arctan$ coming up, but I don't know how to do this ...
1
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0answers
35 views

Question on area under and between curves and volume of a solid by revolution

I have recently begun learning about finding the area under the curve by definite integrals. But I am still a little unsure of the concepts. When you integrate for a certain range of the graph , does ...
1
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1answer
82 views

Second mean value theorem proof

I am asked to prove the second mean value theorem: Let $f$ and $g$ be defined on $[a,b]$ with $g$ continuous, $f\ge 0$, and $f$ integrable. Then there is a point $x_0 \in (a,b)$ such that $$ ...
0
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1answer
26 views

ML-inequality for real integrals

For a homework assignment from my analysis class, I was asked to show the following: Let $f\colon [a,b] \to \mathbb{R} $ be Riemann integrable and $|f(x)| \le M$. Define $F(x) = \int_{a}^{x}f(t)dt$. ...
1
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2answers
94 views

When do you drop the absolute value from ln|x| + C when integrating $\frac{1}{u}du$

Given: p(t) represents the number of cats, when t>=0. Given: p(t) is increasing at a rate directly proportional to $800-p(t)$ So, I represent this as: $\frac{dp}{dt}= k(800-P)$ I want p(t), so I ...
6
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0answers
79 views

Can these two indefinite integrals be evaluated in closed form?

I'm wondering whether any of these two indefinite integrals $$\int \frac{1}{\sqrt{1+\alpha \sinh(x)^{-4/3}}}dx$$ $$\int \frac{\sinh(x)^{-4/3}}{\sqrt{1+\alpha\sinh(x)^{-4/3}}}dx$$ can be evaluated in ...
3
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1answer
120 views

Evaluating $\int_{-\infty}^{\infty}\frac{1}{(x^2+b^2)^2}dx$

Find $$\int_{-\infty}^{\infty}\frac{1}{(x^2+b^2)^2}dx$$ We see that the only poles are at $x=\pm bi$. Integrating over the semicircular contour implies that it is equal to $2\pi i*Res_{(+bi)}$ ...
2
votes
1answer
56 views

Lebesgue point and integration

Let $f$ be in $L^1_{\text{loc}}(\mathbb{R})$. We know that for almost every $t$ $$ \lim_{h\to 0} \frac{1}{h} \int_t^{t+h} |f(u)-f(t)|\text{d} u = 0. $$ My question is : can we say that for almost ...
1
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2answers
54 views

Integral of $((x^2+1)((x-1)^2+1))^{-1}$

Find $$\int_{-\infty}^{\infty}\frac{1}{(x^2+1)(2-2x+x^2)}dx$$ So I am going to integrate this using a semicircular contour. Is it safe to say that on the curved part, the integral vanishes? because ...
2
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0answers
161 views

How to evaluate this integral$\int_{-\infty}^\infty\dfrac{\omega^\alpha e^{i\omega t}}{(\omega_0^2-\omega^2)^{2}+4(\zeta\omega_0\omega)^2}\,d\omega$

How to calculate the following integral? $$\int_{-\infty}^\infty\dfrac{\omega^\alpha e^{i\omega t}}{(\omega_0^2-\omega^2)^{2}+4(\zeta\omega_0\omega)^2}\,d\omega$$ where ...
0
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2answers
28 views

Integration with bounded derivative

I try to solve the following problem, but I have no idea how to link bounded derivative to integral: IN Riemann Sence Let $f:[0,1]\rightarrow \mathbb{R}$ be a differentiable function such that ...
3
votes
1answer
101 views

Contour Integral of $\sin(z)/(z^2-z)$

Find the integral $\int_{\lambda}\frac{\sin(z)}{z(z-1)}$ where $\lambda(t) = 10e^{it},t\in[0,2\pi]$ We notice that there are poles at $z = 0$ and $z=1$. So we can use residue theorem but I am ...
1
vote
1answer
13 views

integration concening Fourier transfom variable and space variable

We define the short time Fourier transform as follows: $$V_{g}f(x,w)=\int_{\mathbb R} f(t)g(t-x)e^{-2\pi itw} dt, (x,w \in \mathbb R).$$ (We may assume that $f$ and $g$ nice functions so that every ...
2
votes
1answer
24 views

Growth of plant in greenhouse

The following problem came up in an exam I sat recently. I got 113cm, but I'm quite unsure about my method. Is someone able to go through the working and explain the problem? Of course, I don't ...
1
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0answers
23 views

Integral on sphere and ellipsoid

Let $a,b,c \in \mathbb{R},$ $\mathbf{A}=\left[\begin{array}{*{20}{c}} \mathbf{a}&{0}&{0}\\ {0}&\mathbf{b}&{0}\\ {0}&{0}&\mathbf{c} \end{array}\right]$ , det A $>1$ Let D = ...
2
votes
1answer
162 views

Show that $u(x)=\ln\left(\ln\left(1+\frac{1}{|x|}\right)\right)$ is in $W^{1,n}(U)$, where $U=B(0,1)\subset\mathbb{R}^n$.

The entire problem statement is: Let $n>1$ and let $U=B(0,1)\subset\mathbb{R}^n$. Show that $u:U\to\mathbb{R}$ given by $$u(x)=\ln\left(\ln\left(1+\frac{1}{|x|}\right)\right)$$ is in ...
0
votes
1answer
95 views

Prove that both iterated integrals exists but $f$ is not integrable

I need to prove that the function $f$, given by: $$f(x,y)= \begin{cases} 1 \iff (x,y) =(\frac p {2^n},\frac q {2^n}): (p,q,n) \in \Bbb N^3, 0<p,q<2^n \\0 \iff (x,y) \neq (\frac p {2^n},\frac q ...
0
votes
1answer
60 views

Indefinite integrals with natural logs [duplicate]

I know the integral of $\frac{1}{x}$ is $\log(x)$ but I'm not sure how to solve this problem, any help would be appreciated: $$ \int^{3}_{2} \frac{1}{x \ln x} $$ I think I need to substitute $x\ln x$ ...