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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2k views

Line Integral, Work in physics

Hi there all: I have a problem! I need to find the work done on a particle that moves from $(0,0)$ to a point $(1,1)$ by a strait line $y=x$. The force acting upon the particle is $F = (y , 2x$). ...
5
votes
3answers
2k views

Find $\int e^{2\theta} \cdot \sin{3\theta} \ d\theta$

I am working on an integration by parts problem that, compared to the student solutions manual, my answer is pretty close. Could someone please point out where I went wrong? Find $\int e^{2\theta} ...
5
votes
3answers
2k views

Upper and lower bound on integral

Consider the following integral $$\int_0^1 (1-x^n)^M \,d x$$ It converges to $0$ as $M\to\infty$, but I would like to find bounds on the convergence rates. What I mean is that it is straightforward ...
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votes
1answer
495 views

Integral without using Euler substitution

Help me please with integral: $$\int \frac{2x-\sqrt{4x^{2}-x+1}}{x-1}\;dx$$ I must solve it without using Euler substitution. Thanks!
5
votes
2answers
216 views

Asking for general form of Integral Inequality of this kind

Let $f\in C^1[0,a]$ and $f(0)=0$. Is it true that $$\int_0^a \left(\sqrt{x}f(x)\right)^{\prime} \left(\frac{f(x)}{\sqrt{x}}\right)^{\prime}\, dx\geq 0\;\;?$$ What is the general form of this type ...
5
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2answers
324 views

Solving integral $\int \frac{\sqrt{1 - x^2} - 1}{x^2 - 1}dx$

I've been asking a lot of integral questions lately. :D This is the integral I'm trying to solve: $$\int \frac{\sqrt{1 - x^2} - 1}{x^2 - 1}dx$$ By replacing $x = \sin(u)$ (thus $dx = \cos(u)du$ and $...
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votes
3answers
1k views

Young inequality

I am trying to prove young's inequality for integrals $$ ab \leq \int\nolimits_0^a \! f(x) \, \mathrm{d}x + \int_0^b \! f^{-1}(x) \, \mathrm{d}x. $$ Can you help me please?
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votes
3answers
485 views

Are there other analytic functions with this property of sinc function?

This question is motivated by my previous post about sinc function. Prove or disprove that $\frac{\sin x}{x}$ is the only nonzero entire (i.e. analytic everywhere) function $f(x)$ on $\mathbb{R}$ ...
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votes
3answers
945 views

Integral of $\frac1{\cos x}$ using t substitution

Okay, so I'm trying to find $ \int \frac1{\cos x}\mathrm{d}x$ using the substitution $t = \tan\left(\frac{x}{2}\right)$. I sub in the trig identity for $\sec$ as $\frac{1+t^2}{1-t^2}$ and then ...
5
votes
3answers
124 views

Is the Riemann integral of a strictly smaller function strictly smaller?

We all know that if $f\leq{}g$ in $[a,b]$ then $$ \int_a^bf\,dx\leq\int_a^bg\,dx $$ now, imagine that we have $f<g$, is it true that $$ \int_a^bf\,dx<\int_a^bg\,dx $$
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votes
2answers
131 views

Why are there conflicting results for integration?

Using my calculator, I had tried to evaluate the definite integral $$\int_{0}^{1}x^x \mathbf{d}x$$ However, according to my CASIO $fx-570$ES, the result was Math ERROR, which led me to believe that ...
5
votes
3answers
168 views

Is there any way to solve integral of $\sqrt{8-x^{2}}$ without using $\sin$ or $\cos$ formulas?

I was thinking about the following integral if I could solve it without using trigonometric formulas. If there is no other way to solve it, could you please explain me why do we replace $x$ with $2\...
5
votes
2answers
162 views

How could one solve $\int_{0}^{\infty} \frac{1}{1-t^4}dt$ with special functions?

How could one solve $$\int_0^\infty \frac{1}{1-t^4} \, dt\,?$$ I have to apply special functions, so I thought that I have to use the change variable $$u=t^4,$$ but $$du=4t^3\,dt$$ and when $$t\...
5
votes
2answers
137 views

How to solve this integral $\int _0^{\infty} e^{-x^3+2x^2+1}\,\mathrm{d}x$

My classmate asked me about this integral:$$\int _0^{\infty} e^{-x^3+2x^2+1}\,\mathrm{d}x$$ but I have no idea how to do it. What's the closed form of it? I guess it may be related to the Airy ...
5
votes
2answers
150 views

How to evaluate an integral from Griffiths

I'm working through Griffith's Electrodynamics book during the winter break, and I'm having trouble on evaluating this integral from problem 2.7 of Introduction to Electrodynamics 4th edition. I have ...
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5answers
100 views

Integrate: $\int\ln(2x+1) \, dx$

$$\int\ln(2x+1) \, dx$$ I setting up this problem and I am finding it hard to understand why $dv= 1$. When using this formula $$\int u\ dv=uv-\int v\ du$$ And using these Guidelines for Selecting ...
5
votes
2answers
75 views

Minimizing $\int_{0}^{1} (1+x^2)f^2(x)dx$

What is $$\min_{f\in D} \int_{0}^{1} (1+x^2)f^2(x)\mathrm dx,$$ where $D$ is the collection of all continuous real functions from $[0,1]$ such that $\int_{0}^{1} f(x)$ = 1. My attempt Note that $\...
5
votes
1answer
70 views

double integral problem $\iint e^{\frac{x}{x+y}}dxdy$

I'm trying to integrate $$\iint e^{\frac{x}{x+y}}dxdy$$ where $y \leq (1-x)$ and $0 \leq x,y \leq 1$. I tried to define new variables as $u=x$ and $v=x+y$, but I can't solve this either. I have ...
5
votes
2answers
241 views

Prove that $\lim_{t\to \infty} t\mu(\{x:f(x)\geq t\})=0$

Problem Suppose $f$ is a non-negative integrable function on a measure space $(X,\mathcal{A},\mu).$ Prove that $$\lim_{t\to \infty} t\mu(\{x:f(x)\geq t\})=0$$ Attempt Let $E_t=\{x:f(x)\geq t\}$ ...
5
votes
3answers
140 views

An improper integral with parameter $\int_{1}^{+\infty}\frac{\ln(1+x^p)}{\sqrt{x^2-1}}$

I have to analyse the convergence of this integral: $$\int_{1}^{+\infty}\frac{\ln(1+x^p)}{\sqrt{x^2-1}}$$ where $p\in \mathbb{R}$. I have thought to write: $$\int_{1}^{c}\frac{\ln(1+x^p)}{\sqrt{x^2-...
5
votes
1answer
149 views

Changing order of integration for the triple integral $ \int\limits_{0}^{2} \int\limits_{0}^{2z} \int\limits_{y}^{2y} f_{(x,y,z)}\; dx\, dy\, dz $

I need to change order of integration for the following triple integral: $$ \int\limits_{0}^{2} \int\limits_{0}^{2z} \int\limits_{y}^{2y} f_{(x,y,z)}\; dx\, dy\, dz $$ The domain of integration is ...
5
votes
3answers
105 views

Integrate $\int \frac{x^5 dx}{\sqrt{1+x^3}}$

I took $1+x^3$ as $t^2$ . I also split $x^5$ as $x^2 .x^3$ . Then I subsituted the differentiated value in in $x^2$ . I put $x^3$ as $1- t^2$ . I am getting the last step as $2/9[\sqrt{1+x^3}x^3 ]$ ...
5
votes
2answers
135 views

evaluate $\int \frac{\tan x}{x^2+1}\:dx$

$$\int \frac{\tan x}{x^2+1}\:dx$$ I used By-parts method setting $u=\tan x$ and $dv=\frac{1}{x^2+1}dx$, but then I got an integral that's more complicated I also thought of trigonometric ...
5
votes
1answer
77 views

Question regarding $\int \tan(x) \sec^2(x) \,dx$

I am asked to find the following but unsure whether or not my solution is valid: $$\int \tan(x) \sec^2(x) \,dx$$ Setting $u=\tan(x)$ and $du=\sec^2(x)\,dx$: $$= \int u\,du$$ $$= \dfrac{u^2}{2}+C$$ $...
5
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1answer
164 views

A difficult one-variable exponential integral

I am trying to work out a closed form for the integral \begin{equation} \int_{0}^{1} \frac{1}{\sqrt{s(1-s)}} \exp\left(-\left(\frac{a}{s} + \frac{b}{1-s}\right) \right) \,ds \end{equation} where $a,b&...
5
votes
4answers
189 views

Evaluating $\int_0^x\frac{\sin(t)}{1+t^2}\mathrm dt$

I cannot seem to figure out how to solve the following problem: $$\int_0^x\frac{\sin(t)}{1+t^2}\mathrm dt$$ I have tried by using integration by parts. I set $u = \sin(t)$, $v = \tan^{-1}(x)$ and ...
5
votes
1answer
135 views

Complex integration $\int_{-\pi}^{\pi} \frac{\sin^2 t}{3+\cos t}dt$

I'm trying to evaluate the integral $$\int_{-\pi}^{\pi} \frac{\sin^2 t}{3+\cos t}dt$$ using complex numbers. Meaning, instead of calculating $$\int_{-\pi}^{\pi} \frac{\sin^2 t}{3+\cos t}dt,$$ I want ...
5
votes
2answers
99 views

How to integrate $\ln \big( b + \sqrt{b^2 + c^2 + x^2}\,\big)$?

I am looking to demonstrate the following result. Any ideas are much appreciated. $$ \begin{align}\int \ln \left( b + \sqrt{b^2 + c^2 + x^2}\right) dx = &\;x \ln \left( b + \sqrt{b^2 +c^2 +x^2}\...
5
votes
1answer
113 views

An advanced integral $\int_0^1 \frac{(2 e)^{-1/y} \left(2 e^{1/y}-e 2^{1/y}\right)}{1-y} \ dy$

I'd like to ask you how you would like to approach the integral below $$\int_0^1 \frac{(2 e)^{-1/y} \left(2 e^{1/y}-e 2^{1/y}\right)}{1-y} \ dy$$ and then recommend me some tools you'd employ. It's ...
5
votes
1answer
86 views

Find an integral with fractions

How to find the integral $$\int_0^\infty \frac{e^{-x^2}}{(x^2+1/2)^2}dx?$$ I find it is difficult to do if I integrate by parts...What's the trick?
5
votes
2answers
169 views

Dogbone countor integral (evalutate $\int_0^1 \frac{x^n}{x^a(1-x)^{1-a}}dx$)

I'm confronted with the following problem which I really don't seem to find a way to solve properly: Let $n\in \mathbb{Z}$ be fixed. Determine for what values of the parameter $a\in\mathbb{C}$ the ...
5
votes
3answers
144 views

Proving convergence of $ \int \limits_0^{\infty} \cos\left(x^2\right) dx $

How would one prove the convergence of $$ \int_0^{\infty} \cos\left(x^2\right) \,\mathrm dx $$ I tried using the integral test for convergence by noting that making the substitution $u = x^2$ means ...
5
votes
3answers
759 views

Are differentiation and integration continuous functions?

Is differentiation a continuous function from $C^1[a,b] \to C[a,b]$? I think it is but I can't prove it... Would it be possible to prove it using theory about closed sets in $C[a,b]$ and their ...
5
votes
2answers
115 views

Evaluation of $\int \frac{x\sin(\sin x)}{x+5} \ dx$

How do we find $$\int \frac{x\sin(\sin x)}{x+5} \ dx\ ,$$ is there any way to take that $\sin x$ out from parent $\sin(\cdot)$ ?
5
votes
1answer
416 views

References on integration: collections of fully worked problems (and explanations) of (1) advanced and (2) unusual techniques

I am searching for two kinds of books. (1) Comprehensive books that collect, explain, and provide many examples (that is, fully worked problems) of advanced integration techniques (that is, ...
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1answer
207 views

How can I evaluate this indefinite integral? $\int\frac{dx}{1+x^8}$

How do I find $\displaystyle\int\dfrac{dx}{1+x^8}$? My friend asked me to find $\displaystyle\int\dfrac{dx}{1+x^{2n}}$ for a positive integer $n$. But looking up I am getting pretty noisy answer for ...
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votes
2answers
117 views

Is $\int^x \cos \frac1t$ differentiable at zero?

From Spivak's Calculus, 4th ed., exc 14-20: Let $$f(x) = \begin{cases} \cos \frac1x, & x\neq 0\\ 0, &x=0. \end{cases}$$ Is the function $\int_0^xf$ differentiable at zero? I'm having ...
5
votes
2answers
272 views

2D Integral of Bessel Function and Gaussians

I've run into the following integral, and I'm not sure how to evaluate it. $$F(k)=\int d^{2}\mathbf{x}\left(e^{-\frac{\left(x-2a\right)^{2}}{4w^{2}}}+e^{-\frac{\left(x+2a\right)^{2}}{4w^{2}}}+2e^{-\...
5
votes
2answers
117 views

Prove $\int\frac{12x\sin^{-1}x}{9x^4+6x^2+1}dx=-\frac{2\sin^{-1}x}{3x^2+1}+\tan^{-1}\left(\frac{2x}{\sqrt{1-x^2}}\right)+C$

How to prove $$\int\frac{12x\sin^{-1}x}{9x^4+6x^2+1}dx=-\frac{2\sin^{-1}x}{3x^2+1}+\tan^{-1}\left(\frac{2x}{\sqrt{1-x^2}}\right)+C$$ where $\sin^{-1}x$ and $\tan^{-1}x$ are inverse of trig functions....
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2answers
83 views

a question about how to prove mutivariable integral, I am struggling about it!

If $f(x)$ is Riemann integrable in $[a,b]$, and then how to prove $$\int_{a}^{b} f(x_1) \, dx_1 \int_{a}^{x_1}f(x_2) \, dx_2 \cdots \int_{a}^{x_{n-1}}f(x_n) \, dx_n={1\over n!} \left[\int_a^b f(x) \, ...
5
votes
2answers
165 views

Proof that $J_{\nu}(x) \sim (x/2)^\nu / \Gamma(\nu+1) \; \text{as} \; \nu \rightarrow \infty$

I'm working through the exercises of Bender and Orszag's famous book, but I got stuck in 6.25 (a), in which it is asked to prove that $$J_\nu (x) \sim (x/2)^\nu / \Gamma(\nu+1) \; \text{as} \; \nu \...
5
votes
2answers
135 views

Evaluate $\displaystyle\int_0^\pi \frac{x}{1+\sin^2x} \ dx$

How can one evaluate $$\int_0^\pi \frac{x}{1+\sin^2x} \ dx\ ?$$
5
votes
1answer
88 views

Please help with integral

Please help me with evaluate the following improper integral $$\int_{0}^{\infty} \frac{\ln (1+u) -\ln 2}{(u+1)\sqrt{u} \ln u} du.$$
5
votes
4answers
174 views

Integrating $ \int \limits_{-\infty}^{\infty} \dfrac{\sin^2(x)}{x^2} \operatorname d\!x $

I'm trying to evaluate $\displaystyle \int \limits_{-\infty}^{\infty} \dfrac{\sin^2(x)}{x^2} \operatorname d\!x $. My first though was to use residue calculus, since we've got the pole of order 2 ...
5
votes
1answer
111 views

$\lim\limits_{n\to\infty}\Bigl[\sin\Bigl(\dfrac{n}{n^2+1^2}\Bigr)+\sin\Bigl(\dfrac{n}{n^2+2^2}\Bigr)…+\sin\Bigl(\dfrac{n}{n^2+n^2}\Bigr)\Bigr]$

Find the limit of $\Bigl[\sin\Bigl(\dfrac{n}{n^2+1^2}\Bigr)+\sin\Bigl(\dfrac{n}{n^2+2^2}\Bigr)...+\sin\Bigl(\dfrac{n}{n^2+n^2}\Bigr)\Bigr]$ using Riemann integrals of a suitable function. $\Bigl[\...
5
votes
1answer
189 views

Limits of Integral $\lim\limits_{n\to\infty}\int\limits_a^b\sqrt[n]{f^n(x)+g^{n}(x)}dx.$

Let $f, g:[a,b]\to[0,\infty)$ be continuous functions. Find the value of $$\lim\limits_{n\to\infty}\int\limits_a^b\sqrt[n]{f^n(x)+g^{n}(x)}dx.$$ One said that the results is $\int\limits_a^b h(x)dx$ ...
5
votes
5answers
106 views

Identity $\int_{-\infty}^{\infty}\frac{e^{uz}}{1+e^u} \mathrm{d}u=\frac{\pi}{\sin(\pi z)}$

I want to prove the identity $$F(z)=\int_{-\infty}^{\infty}\frac{e^{uz}}{1+e^u} \mathrm{d}u=\frac{\pi}{\sin(\pi z)}$$ First of all $F(z)$ defines an analytic function for $0<z<1$. I am little ...
5
votes
1answer
234 views

Show that,$\int_0^\pi \left|\frac{\sin nx}{x}\right|\mathrm{d}x \ge \frac{2}{\pi}\left(1+\frac12+\cdots+\frac{1}{n}\right)$

Show that,$$\int_0^\pi \bigg|\dfrac{\sin nx}{x}\bigg|\mathrm{d}x \ge \dfrac{2}{\pi}\bigg(1+\dfrac12+\cdots+\dfrac{1}{n}\bigg)$$ I could not approach the problem at all. Please help.
5
votes
1answer
182 views

Integral asymptotic expansion of $\int_0^{\pi/2} \exp(-xt^3\cos t)dt$ as $x \to \infty$

I have the integral $$I(x)=\int_0^{\pi/2}\exp(-xt^3\cos t)dt$$ and I want to derive the first two terms in the asymptotic expansion for $x\rightarrow \infty$, which should give me $$\frac{1}{3x^{1/3}}\...
5
votes
2answers
99 views

$f$ integrable $\Leftrightarrow f<\infty$ a.s.?

$f\colon\Omega\to\mathbb{R}$ measurable function on measure space$(\Omega,\mathfrak{A},\mu)$. I am interested to know if then $$ f\text{ is integrable }\Leftrightarrow f\text{ is finite a.s.}~~~. $$ ...