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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11
votes
2answers
182 views

How can I calculate $\int_0^\infty u^3/(e^u-1) \, du$?

How can I calculate $$\int_0^\infty \frac{u^3}{e^u-1} \, du$$ Acutally this is a part of derivation of Stefan-Boltzmann's Law. And equation should give answer $\pi^4/15$.
2
votes
1answer
108 views

Question about $\int_\Omega\!\cos^n\alpha\cdot\cos\theta_o\,d\omega_o$

I see this integral metioned in this paper (at the start of section 3.2 ,p.4) $$\int_\Omega\!\cos^n\alpha\cdot\cos\theta_o\,d\omega_o$$ It's an integral over hemisphere and the $\alpha$ term means ...
3
votes
2answers
175 views

How To Set The Area Between Two Functions Equal To A Constant?

Please pardon the broad nature of this question. Suppose two functions encompass an area between them. What approach might be taken to adjust either function through adding constants to set the area ...
3
votes
1answer
294 views

Definite integral involving modified bessel function of first kind, exponentials, and powers

Is there any solution for this integral? $$ \int_a^b xe^{-\alpha x^2}J_n(\beta x) dx\,. $$ I looked up in all books i could find and only found this: $$ \int_0^{\infty} xe^{-\alpha x^2}J_n(\beta x) ...
5
votes
1answer
118 views

Definite integral, floor function, parameter

Could you tell me how to calculate the following limit? $$\lim_{n \rightarrow \infty} \int_{2\pi n}^{(2n+1)\pi}\!\left(x^t - \left \lfloor x^t \right \rfloor\right) \sin x \,dx$$ for a given ...
0
votes
1answer
59 views

Difference of definite integrals inequality

Could you help me how prove that for any $\mathcal{C}^1$ function we have: $$\left|\int_{a} ^{\frac{a+b}{2}}f(x) d x - \int_{\frac{a+b}{2}} ^bf(x)dx\right| \le \frac{(b-a)^2}{4} \cdot \max _{x \in ...
0
votes
1answer
113 views

Double Integral : changing variables

What i need to compute is the area of a parallelogram doing a variables change from the basic coordinates to the parallelogram coordinates. The corners of the parallelogram are (1,1),(2,3),(5,3) and ...
3
votes
1answer
99 views

Integral inequality using positive and negative parts

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a measurable (with respect to the Lebesgue measure), $\pi$-periodic function which is Lebesgue integrable over $[0,\pi]$. Moreover assume that ...
0
votes
1answer
77 views

How do I calculate the complex integral $\int_c\frac{1}{z^2}\,\mathrm{d}z$?

How do I calculate the complex integral $\displaystyle\int_c\frac{1}{z^2}\,\mathrm{d}z$, where $c$ is a direct line from $z=0$ to $z=1+2i$? Is it possible solving such the integral? I mean ...
0
votes
2answers
32 views

Solving the next Integral (by parts?)

Can anyone give me a hint / help with the next integral? Thanks! $$\displaystyle\int_{0}^{t}{x^{a-1}(t-x)^{b-1}dx}$$
0
votes
1answer
106 views

How can I evaluate $\int_0^\infty \frac{\sin x}{x} \,dx$? [may be duplicated] [duplicate]

How can I evaluate $\displaystyle\int_0^\infty \frac{\sin x}{x} \, dx$? (Let $\displaystyle \frac{\sin0}{0}=1$.) I proved that this integral exists by Cauchy's sequence. However I can't evaluate ...
1
vote
1answer
336 views

Lebesgue vs. Riemann integrable function

While trying to learn the difference between Lebesgue and Riemann integrals, I came across the following example: $$\int_{0}^{1}t^\lambda\,\mathrm dt$$ What I know so far: only for $\lambda>0$ ...
0
votes
1answer
55 views

Does Simpsons rule still apply when a < 0?

I am currently working on an assignment where I have to find the answer to the following integral using Simpsons rule:$\int x+1$ (MIN = -1 MAX = 3), I choose to have 6 intervals. I now start ...
3
votes
2answers
59 views

integrate product of trig functions

I need to find the Fourier cosine series for $\cos(3x)\sin^2(x)$, But I don't even know where to start to determine $$\int _0^{\pi }\cos(3x)\sin^2(x)\cos(k x)dx$$
5
votes
2answers
174 views

Integrate: $\int \frac{dx}{x \sqrt{(x+a) ^2- b^2}}$

How to evaluate $$\int \frac{dx}{x \sqrt{(x+a) ^2- b^2}}$$ I tried trigonometric substitution $x + a = b \sec \theta$ and I encountered $$\int \frac{\tan \theta}{ (b - a \cos \theta) \sqrt{\tan^2 ...
1
vote
6answers
235 views

Find the limit : $\lim\limits_{n\rightarrow\infty}\int_{n}^{n+7}\frac{\sin{x}}{x}\,\mathrm dx$

I have this exercise I don't know how to approach : Find the limit : $$\lim_{n\rightarrow\infty}\int_{n}^{n+7}\frac{\sin x}{x}\,\mathrm dx$$ I can see that with $n\rightarrow\infty$ the area ...
2
votes
1answer
170 views

prove this $\int_{0}^{2}f^2(x)dx\le\int_{0}^{2}f'^2(x)dx$

let $f\in C^1[0,2]$,and such $\int_{0}^{2}f(x)dx=0,f(0)=f(2)$, show that $$\int_{0}^{2}f^2(x)dx\le\int_{0}^{2}f'^2(x)dx$$ I think we must use $Cauchy$ inequality my idea:I have see this let ...
1
vote
1answer
88 views

If $\int^{1}_{0} \frac{\tan^{-1}x}{x} dx = k \int^{\pi/2}_{0} \frac {x}{\sin x} dx$, find $k$.

Problem: If $$\int^{1}_{0} \frac{\tan^{-1} x}{x} dx = k \int^{\pi/2}_{0} \frac{x}{\sin x} dx,$$ find $k$. Solution: If we put $x=\tan t$ in $$\int^{1}_{0} \frac{\tan^{-1}x}{x} dx$$ then ...
2
votes
1answer
101 views

The notion of a curve in the context of line integrals

For brevity I'm making the following assumption: I'm only talking about regular curves on $\left[a,b\right]$ with values in $\mathbb{R}^{n}$, and line integrals of scalar fields. [Since there are a ...
2
votes
0answers
183 views

How to calculate this integral containing a Dirac delta function

I need to calculate this integral: $$u(x,t)=\frac{1}{4\pi c} \int_{-\infty}^{\infty} \frac{1}{\rho_1} \delta(\rho_1 - c(t - t_1)) dz_0$$ In this integral $$ \rho_1 = \sqrt{(x-x_1)^2 + (y-y_1)^2 + ...
5
votes
0answers
177 views

A difficult integral $\int_0^\infty \mathrm{d}t\frac{1}{t}\frac{1}{t-s-\mathrm{i}\epsilon}\frac{1}{X}\ln\frac{1-X}{1+X} $

Can anyone give any hints on how to rewrite this in terms of dilogarithms? $$\int_0^\infty ...
2
votes
4answers
98 views

For what values of $a$ $\int^2_0 \min(x,a)dx=1$

I want to check for what values of $a$ the integral of this function is equal to $1$ $$\int^2_0 \min(x,a)dx=1$$ What I did is to check 2 cases and I am not sure is enough : case 1: $$a<x ...
5
votes
1answer
104 views

How find this $\int_{0}^{\infty}xe^{-x}\left(\int_{0}^{\pi/2}(1-e^{x-x\csc{t}})\sec^2{\!t}\,\mathrm dt\right)^2\,\mathrm dx=\frac{1}{3}$

This problem was taken from here: Show that $$\int_{0}^{\infty}xe^{-x}\left(\int_{0}^{\pi/2}(1-e^{x-x\csc{t}})\sec^2{\!t}\,\mathrm dt\right)^2\,\mathrm dx=\dfrac{1}{3}$$ my idea: ...
4
votes
2answers
80 views

Problem in computing of integral by substitution.

I want to compute an integral like $\int_0^{+\infty} \ln(1+x)e^{-x}\,\mathrm dx$. Then denote $\mu = 1-e^{-x}$, so $x=-\ln(1-\mu)$. Substitute this into the integral, we get $$\int_0^1 ...
3
votes
1answer
63 views

What is meant by $|dxdy|^{1/2}$ in the integral?

In this Daniel Grieser - Basics of the b-calculus paper the author mentions the term of a half-density on page 54 as an object which look like $u(x) |dx_1 \cdots dx_n|^{\frac{1}{2}}$. And I'm not ...
14
votes
4answers
230 views

Compute $\int_{0}^{1}\left[\frac{2}{x}\right]-2\left[\frac{1}{x}\right]dx$

The question is to find $$\int_{0}^{1}\left[\dfrac{2}{x}\right]-2\left[\dfrac{1}{x}\right]dx,$$ where $[x]$ is the largest integer no greater than $x$, such as $[2.1]=2, \;[2.7]=2,\; [-0,1]=-1.$ Is ...
9
votes
1answer
240 views

Does this integral have a closed form: $\int_0^1 \frac{x^{\beta-1}dx}{1-x}\log\frac{1-y x^\delta}{1-y}$?

Consider the following integral: $$G(\beta,\delta,y) = \int_0^1 \frac{x^{\beta-1}dx}{1-x}\log\frac{1-y x^\delta}{1-y},$$ with $\delta>0$, $\Re\beta>0$, $y\neq1$. Does it have a closed form in ...
1
vote
1answer
74 views

Differentiable functions without an antiderivative

Specifically, why is there no antiderivative, or any possible method of integrating (except numerically) say $\;e^{\csc(x)}$? (I don't have my computer handy right now so I cant format the formula, ...
13
votes
2answers
278 views

A new constant?

I was experimenting in Wolfram Alpha the answer to the equation $\int_0^k x^x dx=1$ And I got about 1.19... But, What is this number k (and could you calculate it to more decimal places?) And is it ...
2
votes
2answers
244 views

Incomplete beta function in MATLAB

Mathematica is capable of evaluating the incomplete beta function $$ \mathrm{Beta}[z,a,b] = \int_0^z u^{a-1}\left(1-u\right)^{b-1}\,du $$ even when the argument $z$ is negative. MATLAB's function ...
5
votes
1answer
101 views

General Integral Formula

I know how to find the integral below, but I would like to know if there is any clever or general formula for the integral, since my method involves simple polynomial division... $\int ...
2
votes
0answers
50 views

Generalized Riemann Integral

Is there any usage of studying the Henstock-Kurzweil integral as such ? It doesn't seem to be as popular a method of integration as the Lebesgue integral or even the Riemann-Stieltjes ...
4
votes
2answers
134 views

Limit of product of $\sin \frac{k}{n}$

Could you help me how to find the limit of $$\left(\sin \frac{1}{n} \cdot \sin \frac{2}{n} \cdots \sin 1\right)^{\frac{1}{n}}?$$ I know that $$\ln \left((\sin \frac{1}{n} \cdot \sin \frac{2}{n} ...
2
votes
2answers
90 views

$f_{n+1}(x):= \int_a ^x f_n(t)dt$, $\sum_{m=1} ^{\infty} f_m(x)$ is uniformly convergent

Let $f_1 : [a,b] \rightarrow \mathbb{R}$ be an integrable function. Let's define a sequence $(f_n)$, $ \ \ f_n : [a,b] \rightarrow \mathbb{R}$ as $f_{n+1}(x):= \int_a ^x f_n(t)dt$.' Prove that ...
3
votes
1answer
93 views

For a continuous function $f (t ), 0 ≤ t ≤1,$ the integral equation…

I am stuck with the following problem: For a continuous function $f (t ), 0 ≤ t ≤1,$ the integral equation $y(t)=f(t)+3 \displaystyle \int_{0}^{1}tsy(s)ds \,$ has (a) a unique solution ...
10
votes
4answers
383 views

Evaluate the integral $\int_{0}^{\infty} \frac{1}{(1+x^2)\cosh{(ax)}}dx$

The problem is : Evaluate the integral $$\int_{0}^{\infty} \frac{1}{(1+x^2)\cosh{(ax)}}dx$$ I have tried expand $\frac{1}{\cosh{ax}}$ and give the result in the following way: First, note ...
1
vote
3answers
51 views

Need help with $\int \dfrac{2x}{4x^2+1}$

We want$$\int \dfrac{2x}{4x^2+1}$$ I only know that $\ln(4x^2 + 1)$ would have to be in the mix, but what am I supposed to do with the $2x$ in the numerator?
1
vote
3answers
93 views

Integrals using Arctangens

We want to find $\displaystyle \int\dfrac{12}{16x^2 +1}$ I rewrote it to the form $ 3 \cdot \dfrac{1}{u^2 + 1} \cdot u' $ where $u=4x$. I found out that the correction sheet does the same thing, but ...
1
vote
2answers
57 views

Arctangent integral

How come this is correct: $$\int \dfrac{3}{(3x)^2 + 1} dx = \arctan (3x) + C$$ I learned that $$\int \dfrac{1}{x^2+1} = \arctan(x) + C$$ But I don't see how you can get the above one from the ...
1
vote
1answer
118 views

How to solve integrals of type $ \int\frac{1}{(a+b\sin x)^4}dx$ and $\int\frac{1}{(a+b\cos x)^4}dx$

$$\displaystyle \int\frac{1}{(a+b\sin x)^4}dx,~~~~\text{and}~~~~\displaystyle \int\frac{1}{(a+b\cos x)^4}dx,$$ although i have tried using Trg. substution. but nothing get
2
votes
1answer
95 views

How can I devise a general approach to solving an indefinite integration problem?

INTRODUCTION: I'm trying to create a general approach in solving integration problems around 7 specific methods. Basic Formulas, Substitution, Numerical Integration, Integration by Parts, ...
5
votes
3answers
764 views

Volume integral of the curl of a vector field

I am having hard time recalling some of the theorems of vector calculus. I want to calculate the volume integral of the curl of a vector field, which would give a vector as the answer. Is there any ...
0
votes
0answers
31 views

About Stieltjes Sums

Let $[a,b]\subset \mathbb{R}$. A tagged partition of $[a,b]$ is a set $D=\{(t_i,[x_{i-1},x_i])\}_{i=1}^m$ where $\{[x_{i-1},x_i]\}_{i=1}^m$ is a partition of $[a,b]$ and $t_i\in [x_{i-1},x_i]$; $t_i$ ...
3
votes
3answers
137 views

Uniform convergence of composition of functions and integration

I've been wondering about this problem for a bit, it came up in class but its not really homework. Let $f:[0,1]->R$ be continuous and non-negative. We know $f(x^n)\to f(0)$ for $x \in [0,1)$ and ...
4
votes
2answers
193 views

Evaluate : $\int^{\frac{\pi}{2}}_0 \frac{\cos^2x\,dx}{\cos^2x+4\sin^2x}$

Evaluate: $$\int^{\frac{\pi}{2}}_0 \frac{\cos^2x\,dx}{\cos^2x+4\sin^2x}$$ First approach : $$\int^{\frac{\pi}{2}}_0 \frac{\cos^2x\,dx}{\cos^2x+4(1-\cos^2x)}$$ $$=\int^{\frac{\pi}{2}}_0 ...
15
votes
4answers
1k views

$\int^{1}_{0} f^{-1} = 1 - \int^1_0 f$

One more from hard to believe facts, which I'm curious why are true. Let $f : [0,1] \rightarrow [0,1] $ is a continuous, monotonically increasing and surjective function Then $$\int^{1}_{0} f^{-1} ...
3
votes
2answers
96 views

How to prove that $\int^1_0 \frac{1}{x^x} dx = \sum^{\infty}_{n=1} \frac{1}{n^n} $? [duplicate]

The task is: Prove, that $$\int^1_0 \frac{1}{x^x} dx = \sum^{\infty}_{n=1} \frac{1}{n^n}$$ I completly don't have an idea, how to prove it. It seems very interesting, I will be glad if someone ...
3
votes
1answer
59 views

Integration by substitution problem $x = C \sin(t)$.

For solving the integral: $$ \int_a^b \sqrt{\alpha^2 - \beta^2 x^2} \, dx $$ I've been taught to use $x = \frac{\alpha}{\beta} \sin(t)$ in order to get $$ \frac{\alpha^2}{\beta} \int_{\arcsin(a ...
2
votes
2answers
86 views

Finding indefinite integral by partial fractions

$$\displaystyle \int{dx\over{x(x^4-1)}}$$ Can this integral be calculated using the Partial Fractions method.
4
votes
1answer
103 views

How to find the following indefinite integral?

$$ \int {dx \over {\sin^3 x+\cos^3 x}}$$ Can this integral be found by substitution or any other method such as complex number?