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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113 views

Let $f>0$ differentiable in $[0,\infty)$. Assume $\lim \limits_{x \to \infty} (\log\circ f)^\prime(x) < 0$. Show that $\int_0^\infty f$ converges.

So what I gathered from the givens about $f$, since $(\log\circ f)^\prime(x)=\frac{f^\prime(x)}{f(x)}$ it would mean that far enough, $f^\prime(x)<0$. I don't know how to go about this from here. ...
4
votes
2answers
96 views

Bounding $I(x)=\int_{0}^{x}\frac{(x-t)^2\exp(t)}{2}\mathrm dt$

Suppose $\forall x \in \mathbb{R}$, $I(x)=\displaystyle\int_{0}^{x}{\dfrac{(x-t)^2\exp(t)}{2}\,\mathrm dt}$. Without calculating $I(x)$ how can I prove that: $\forall x \ge 0$, $\quad0\le I(x) \le ...
4
votes
2answers
1k views

Evaluate $\int\frac{1}{1+(\tan x)^\sqrt2}\ dx$

How can we evaluate $$\int\frac{1}{1+(\tan x)^\sqrt2}\ dx$$ Can you keep this at Calculus 1 level please? Please include a full solution if possible. I tried this every way I knew and I couldn't get ...
4
votes
2answers
657 views

Integration by parts of delta function

I am having the worst time trying to solve this integral, $$ \int g(t)\frac{d}{df}\delta[f(t)]dt, $$ $$ = \int g(t)\frac{dt}{df}\frac{d}{df}\delta[f(t)]dt. $$ This should yield, $$ -\bigg[ ...
4
votes
3answers
98 views

Convergence of $\int_3^{\infty} \frac{1}{(\ln(x))^2(x-\ln(x))}$

Does this integral converge? $$\int_3^{\infty} \frac{1}{(\ln(x))^2(x-\ln(x))}$$ I've been trying to solve this for the past 2 hours...literally. I know the answer is fairly simple, but I just can't ...
4
votes
2answers
301 views

$3\int_{0}^{1}(f'(x))^2dx \geq (2\int_{0}^{1}f(x)dx)^2 \impliedby 2\int_{0}^{\frac{1}{2}}f(x)\,\mathrm dx=\int_{\frac{1}{2}}^{1}f(x) \,\mathrm dx$

Let $f : \mathbb{R} \to \mathbb{R} $ be a differentiable function. Suppose that $2\int_{0}^{\frac{1}{2}}f(x)\,\mathrm dx=\int_{\frac{1}{2}}^{1}f(x) \,\mathrm dx$ Show that $$3\int_{0}^{1}(f'(x))^2 ...
4
votes
5answers
305 views

Integration by parts of $\int_0^\infty \! n^2 \ln(1-e^{-an}) \, \mathrm{d} n$

Im trying to do the following integration by hand, $\int_0^\infty \! n^2 \ln(1-e^{-an}) \, \mathrm{d} n$ I have tried to use integration by parts and substitution, but each time it gets to ...
4
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3answers
1k views

Binomial distribution with Uniform parameter

I have a problem with following exercise (it comes from Geoffrey G. Grimmett, David R. Stirzaker, Probability and Random Processes, Oxford University Press 2001, page 155, ex. 6): Let $X$ have the ...
4
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1answer
93 views

Is this function positive?

I was wondering if: $$\int_0^1x(t)\int_0^tx(s)ds\ dt$$ is positive for a general $x\in L_2[0,1]$ . Can you help me with this?
4
votes
3answers
3k views

Some way to integrate $\sin(x^2)$?

Because the straight forward approach involves Fresnel integrals I thought about a different approach of taking the imaginary part of $\int_{-\infty}^{\infty}\exp(ix^2) $ but have no idea how to ...
4
votes
3answers
262 views

Evaluate $\lim_{n \to \infty}\int^n_1 \frac{\left |\sin x \right |}{n}dx$

Evaluate $$\lim_{n \to \infty}\int^n_1 \frac{|\sin x|}{n}dx$$ I think that I should deal with $\int|\sin x|dx$, but I don't know how to go on. Please help. Thank you.
4
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3answers
324 views

A discontinuous integral

I got curious about a Mathoverflow question and so I read about the so called explicit formula about the zeta function in Davenport's book in analytic number theory. Everything looks good to me except ...
4
votes
2answers
1k views

What is the integral of 1/x?

What is the integral of $1/x$? Do you get $\ln(x)$ or $\ln|x|$? In general, does integrating $f'(x)/f(x)$ give $\ln(f(x))$ or $\ln|f(x)|$? Also, what is the derivative of $|f(x)|$? Is it $f'(x)$ or ...
4
votes
2answers
163 views

Compute: $\lim_{n\to\infty} n^{p+1} \int_{0}^{1} e^{-nx} \ln (1+x^p) \space dx $

Find the following limit for any $p$ natural number: $$\lim_{n\to\infty} n^{p+1} \int_{0}^{1} e^{-nx} \ln (1+x^p) \space dx $$ If i'm not wrong, without much effort one may see that this integral ...
4
votes
2answers
798 views

Integral of $x^2\ln(x)$ using Simpson's rule

This is my homework question: Calculate $\int_{0}^{1}x^2\ln(x) dx$ using Simpson's formula. Maximum error should be $1/2\times10^{-4}$ For solving the problem, I need to calculate fourth derivative ...
4
votes
1answer
223 views

Is there a “continuous product”?

Is there a "continuous product" which is the limit of the discrete product $\Pi$, just like the integral $\int$ is the limit of summation $\sum$. Thanks!
4
votes
2answers
904 views

Help calculating $\int_C e^{-1/z}\sin(1/z)dz$ over the unit circle?

I found the integral $$ \int_C e^{-1/z}\sin(1/z)dz $$ over the circle $|z|=1$ while doing some problems in Schaum's Outline for Complex Variables. This integral has me stumped. The answer is $2\pi ...
4
votes
5answers
320 views

$f(x)$ is positive, continuous, monotone and integrable in (0,1]. Is $\lim_{x \rightarrow 0} xf(x) = 0$?

I'm having trouble with this question from an example test. We have a positive function $f(x)$ that's monotone, continuous and integrable in $(0,1]$. Is $\lim_{x \rightarrow 0} xf(x) = 0$? Progress ...
4
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1answer
117 views

Notation for some integrals

I've seen some problems where the OP writes integrals in this form $$\int {dt} f\left( t \right)$$ or for double integrals $$\int {dx} \int {dtf\left( {t,x} \right)} $$ Do they represent another ...
4
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1answer
123 views

How to evaluate $\int \frac {xe^2}{(1+2x)^2}dx$

I stack with following question. $\int \frac {xe^2}{(1+2x)^2}dx$ I think I need to use $uv-\int vdu$ to evaluate this function but I couldn't see which would be $u$ and $v$ If you have any idea ...
4
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1answer
113 views

Inequality for the integral $\frac{\ln x}{x^n}$

Define the integral $I_{n}$ as follows for $n$ an integer greater than $1$: $I_{n}:=\int_{1}^{e}\frac{\ln x}{x^n}dx$ Is it true that $$I_{n}\leq \frac{1}{n-1}\left(1-\frac{1}{e^{n-1}}\right)?$$ ...
4
votes
4answers
337 views

How to think about derivatives in an abstract fashion?

Derivatives seem easy to understand abstractly as the rate of change of something, higher order derivatives are the rate of change of the rate of change of something, and so on. I, however, have ...
4
votes
1answer
2k views

How to derive the Ornstein-Uhlenbeck Stochastic Integral Equation?

I have a question regarding the Ornstein -Uhlenbeck process. We have a simplified version with Stochastic Integral Equation: $X_t=-a\int^t_0 X_s\,ds +B_t$. B is the Brownian motion. And its analytic ...
4
votes
1answer
145 views

How to estimate this integral

How to estimate the following integral: $$\int_e^x \log{\log{t}} dt$$ so that the error term is within $$O\left(\frac{x}{\log^2{x}}\right)$$. Assume $$x>e$$ Any hint?
4
votes
4answers
207 views

Assigning value to divergent integral

I have an integral of the form $$\int^\infty_{-\infty}\mathrm d \omega \, \frac{\omega^2}{k^2 + \gamma^2 \omega^2}$$ which diverges. This integral should have a finite value, as it must related to ...
4
votes
2answers
974 views

Expectation of pairwise differences between uniform random points in hypercube

Say you have 2 iid random variables $x,y\sim U[0,1]^k$, i.e. the uniform distribution over the k-dimensional unit cube. What's the expected value of the Euclidean distance between them when they have ...
4
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2answers
409 views

Can $\int_0^{\infty} e^{-a^2 t^2 - b t} \sin c t \mathrm dt$ be done?

I'm a physicist and I've been encountering integrals like $\int_0^{\infty} e^{-a^2 t^2 - b t} \sin c t \;\mathrm dt$, where everything is real. Mathematica could not solve it and I could not find it ...
4
votes
3answers
2k views

Is there a closed form for $\int x^n e^{cx}\,\mathrm dx$?

Wikipedia gives this evaluation: $$ \int x^ne^{cx}\,\mathrm dx=\frac1cx^ne^{cx}-\frac nc\int x^{n-1}e^{cx}\,\mathrm dx=\left(\frac{\partial}{\partial c}\right)^n\frac{e^{cx}}{c}$$ But I have no idea ...
4
votes
1answer
99 views

Why isn't this simpler partition enough?

I'm trying to figure out a lecture example given on our Analysis course. We are currently going through Riemann integrals. Let $g:[0,1] \to R, g(x) = 1$ when $x \in [0, \frac{1}{2}]$ and $g(x) = 2$ ...
4
votes
2answers
465 views

These calculations are correct ? About $\int\frac{e^{-x}}{x}dx$

Was trying to calculate $$\int_{0}^{\infty}e^{-x}\ln x dx=-\gamma$$ and I found this question: I want to analyze $$\int\frac{e^{-x}}{x}dx$$ With $u=\displaystyle\frac{1}{x} \Rightarrow du = ...
4
votes
1answer
713 views

Approximating $\pi$ using Monte Carlo integration

I'm trying to approximate $\pi$ using Monte Carlo integration; I am approximating the integral $$\int\limits_0^1\!\frac{4}{1+x^2}\;\mathrm{d}x=\pi$$ This is working fine, and so is estimating the ...
4
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1answer
40 views

Simple Derivation of Functional Equation Question (L'Hospital's Rule)

First, the question is: $f$ is a differentiable function and $f : R \rightarrow R$ $xf(x)-yf(y)=(x-y)f(x+y)$ $f'(2x)=?$ My approach for problem is using L'Hospital's rule: $$ ...
4
votes
2answers
56 views

compute improper integrals using integration by parts

Compute \begin{equation*} \int_0^\infty \frac{\sin^4(x)}{x^2}~dx\text{ and }\int_0^\infty \frac{\sin (ax) \cos (bx)}{x}~dx. \end{equation*} For the first integral I tried letting $u = \sin ^4 x$ ...
4
votes
3answers
56 views

How can I solve the integral $ \int {1 \over {x(x+1)(x-2)}}dx$ using partial fractions?

$$ \int {1 \over {x(x+1)(x-2)}}dx$$ $$ \int {A \over x}+{B \over x+1}+{C \over x-2}dx $$ I then simplified out and got: $$1= x^2(A+B+C) +x(C-2B-A) -2A$$ $$A+B+C=0$$ $$C-2B-A=0$$ $$A=-{1 \over 2}$$ ...
4
votes
1answer
100 views

Integral of a square compared to the square of an integral

What can be said about a complex valued, continuous function $f$, defined on $[0,1]$, such that: $$ \int_0^1{|f|^2}=\left|\int_0^1{f}\right|^2 $$ I encountered this form as part of an exercise. ...
4
votes
2answers
65 views

Integrate gaussian times sqare root of x times polynomial of order 2

How does one evaluate integrals like: $$ \int_0^\infty{\sqrt{x}\,\dfrac{\left(x-a\right)^2}{2\sigma^2}\,\exp{\left(-\dfrac{\left(x-\mu\right)^2}{2\sigma^2}\right)}}\;dx $$ There is a real and finite ...
4
votes
1answer
69 views

Evaluate $\int_{-1}^1 \frac {1}{\sqrt{|x|}} \text{d}x$

I need some help to solve this integral with absolute value. I'm not sure how to do these types of integrals. $$\int_{-1}^1 \frac {1}{\sqrt{|x|}} \text{d}x$$ Thank you
4
votes
4answers
65 views

$\frac{d}{dx} \int_{a}^{x} f(x,t) \ dt$

Does $$\frac{d}{dx} \int_{a}^{x} f(x,t) \ dt$$ equal to $f(x,x)$ by Fundamental Theorem of Calculus?
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2answers
50 views

Help to understand this property: $\int\limits_{ka}^{kb}s\left(\frac{x}{k}\right)dx = k\int\limits_a^bs(x)dx$

I'm reading the part explaining the properties of the integral of a step function in Apostol's Calculus I and he explains this property: $$\int\limits_{ka}^{kb}s\left(\frac{x}{k}\right)dx = ...
4
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2answers
77 views

Indefinite integral of $\arctan{\sqrt{1-x^{2}}}$

All is in the title: what is the antiderivative of $x\mapsto \arctan{\sqrt{1-x^{2}}}$ ? I'm supposed to tutor younger students taking an integration class, and this is one of their exercises. I ...
4
votes
2answers
31 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 = ...
4
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2answers
126 views

Approximation for elliptic integral of second kind

My (physics) book gives the following approximation: $\int_{-\pi/2}^{\pi/2} \sqrt{1-(1-a^2) \sin(k)^2} dk \approx 2 + (a_1 - b_1 \ln a^2) a^2 + O(a^2 \ln a^2)$ where a1 and b1 are "(unspecified) ...
4
votes
3answers
102 views

Convergence of $\int_0^\infty $sin$ (x^p) dx$

Consider the $\displaystyle \int_0^\infty $sin$ (x^p) dx$. Does it converge when $p<0$? Does it converge when $p>1$? My Work: Let $x^p=y$, then $\displaystyle \int_0^\infty $sin$ ...
4
votes
2answers
155 views

Integral along a contour is $0$, how?

I recently had an extremely failed attempt at asking the same question, so I am posting the same question more or less to hope that someone can give me feedback. Consider the integral: ...
4
votes
3answers
150 views

Residue Theorem for Gamma Function

I am kinda stuck and not sure what to do at this point of the calculation where: $$\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left(\,x^{-1}\sigma\,\sqrt{\, 2\,}\,\,\right)^{s}\Gamma\left(\,{s \over ...
4
votes
1answer
72 views

Reduction formula of primitive $\big(1-\sin^3{x}\big)^n\cos{x}$

I am trying to obtain a reduction formula for $$\int_0^{\pi/2}\big(1-\sin^3{x}\big)^n\cos{x}\;\mathrm dx $$ where $n \in \mathbb{N}$. My attempt is as follows $$\text{let } v = \sin{x}\; \implies ...
4
votes
1answer
105 views

Evaluating $\int_0^{\infty} \frac{\sqrt{x}}{x^2+2x+5} dx$ using complex analysis

how do I compute $$\int_0^{\infty} \frac{\sqrt{x}}{x^2+2x+5} dx$$ with complex analysis? I feel like im calculating the residue wrong and I cant get to the answer correctly. I tried to branch cut ...
4
votes
3answers
63 views

Evaluate $\iint_Rxy^2\sqrt{x^2+y^2}dxdy$, where $R={(xy)\in{\Bbb{R}^2}:1\le{x^2}+y^2\le{4},y\ge{0}}.$

I have no idea where to start with this one? And how to step through it. How would I set the limits for this? And then what would I go on to do? Thanks!
4
votes
2answers
197 views

Finding the area of a implicit relation

Let's say we have the function: $$x^2+y^2+\sin(4x)+\sin(4y)=4$$ I haven't taken Calculus III, in fact I'm just taking Calculus I. Since I learned how to find the derivative of implicit relations I ...
4
votes
2answers
174 views

Computing $\sum_{n=1}^{\infty} \frac{\psi\left(\frac{n+1}{2}\right)}{ \binom{2n}{n}}$

Here is an interesting series I played with, namely $$\sum_{n=1}^{\infty} \frac{\displaystyle\psi\left(\frac{n+1}{2}\right)}{\displaystyle \binom{2n}{n}} \approx -0.245969181104090562617616399148$$ ...