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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9
votes
0answers
321 views

Integral $\int_0^\infty \frac{\log^2 x \cos ax}{x^n-1}dx$

Hi I am trying to calculate $$ I:=\int\limits_0^\infty \frac{\log^2 x \cos (ax)}{x^n-1}\mathrm dx,\quad \Re(n)>1, \, a\in \mathbb{R}. $$ Note if we set $a=0$ we get a similar integral given by $$ ...
4
votes
0answers
206 views

Integral $I=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx$

Hi I am trying to integrate and obtain a closed form result for $$ I:=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx. $$ Here is what I tried (but I do not think this is ...
2
votes
1answer
54 views

Integrating Differential Forms

This is part of a homework problem. I want to actually solve it myself, so no solutions, please (although this isn't even the full problem statement). I don't have a very good grasp on differential ...
12
votes
2answers
282 views

Area interpretation of integrals

When integrating under part of a circle, as in $$A=\int_0^a {\sqrt{r^2-x^2}\,\mathrm{d}x}$$ I noted that the simple geometric solution would be to add the areas of the sector and triangle formed by ...
3
votes
1answer
51 views

Sum as an integral

Recently I have encountered weird notation that I don't see into. When I have some infinite sum $$\sum_{n=1}^{\infty}f(n)$$ I would rewrite it without thinking to the integral form like this ...
0
votes
0answers
155 views

Definite integral of greatest integer function

I need to find the area under a function modeled by $f(x)=\left\lfloor 2.4x \right\rfloor+5$. I can't seem to figure out what the antiderivative of this is, so I'm going try to use a right Riemann ...
1
vote
1answer
48 views

Let $\int_a^bf(x)sgn(f(x)) + 2f(x) \ dx = 0$. Show that $f$ has at least one root.

The Assignment: Let $a,b \in\mathbb{R}$ and $a < b$. Furthermore let $f: [a,b] \rightarrow \mathbb{R}$ be differentiable and $|f(x)| + |f'(x)| \neq 0$ for $\forall x \in [a,b]$. Now, let ...
1
vote
2answers
36 views

An integral involving two variables and the floor function

Let $N$ be some fixed positive integer. I have the following function $$ g(z) = z \int_1^N [t] e^{2 \pi i t z} \ dt. $$ How would one compute $$ \int_0^1 g(z) \ dz ? $$ Thanks!
0
votes
1answer
36 views

Limits of integration

Is there any difference between the $$\int_a^b f(x) dx $$ and $$\lim_{x\to b^-} \int_a^x f(x) dx \qquad \text{OR}\qquad \lim_{x\to a^+} \int_x^b f(x) dx$$ When would one need the second versions of ...
0
votes
3answers
96 views

Evaluate $\displaystyle\int \frac{1}{1+x}\, dx$

I forgot about integrals so I need some help in this problem $\displaystyle\int \frac{1}{1+x}\, dx$ please.
0
votes
1answer
61 views

Showing that this function is not riemann integrable.

Consider the function h defined by h(x) := x+1 for x an element of [0,1] rational, and h(x) := 0 for x an element of [0,1] irrational. Show that h is not Riemann integrable. The hint in the back of ...
1
vote
3answers
97 views

Show $\displaystyle\int_0^af(x)g(x)dx\ge\int_0^af(a-x)g(x)dx$

Assume $f$ and $g$ are monotonically increasing on $[0,a]$, Show that $$\displaystyle\int_0^af(x)g(x)dx\ge\int_0^af(a-x)g(x)dx$$ If I differentiate both sides w.r. to $a$ then; ...
2
votes
2answers
19 views

Finding an integral $\int g(x)^j dx $ from $\int g(x)^2 dx $

let $I = \int_0^1 g(x)^2 dx $, where $g$ is a real valued function. With this information is it possible to give an upper bound for $\int_0^1 g(x)^j dx $? Here $j$ is a natural number. When $j=1$ I ...
3
votes
2answers
70 views

How to show that $f$ is Riemann integrable

Let $u,v:[a,b]\rightarrow\mathbb{R}$ be contunious. Define $f:[a,b]\rightarrow\mathbb{R}$ by $$f(x) = \begin{cases}u(x) & x \in \mathbb{Q} \\ v(x) & x \in \mathbb{R}-\mathbb{Q}\end{cases}$$ ...
2
votes
1answer
98 views

How to integrate $\displaystyle\int_o^\pi\frac{dx}{\sqrt{3-\cos(x)}}$?

How to integrate $\displaystyle\int_o^\pi\frac{dx}{\sqrt{3-\cos(x)}}$ ? If I take $y=\sin\left(\frac{x}{2}\right)$ then, $\displaystyle ...
0
votes
1answer
66 views

Definition of the integral of a vector field on Riemannian manifold and Euclidean spaces

Given a compact Riemannian manifold $(M,g)$ and a vector field $X \in \mathfrak{X}(M)$, is it possible to define the integral of $X$ on $M$? What if $M$ is a Euclidean space? Clearly the definition ...
1
vote
0answers
45 views

Turning a summation into an integral

I have a summation of the form: $$y(x) = \sum\limits_{h=-L}^L\frac{A(h)\cdot R(h)^2}{((x-h)^2+R(h)^2)^{3/2}}$$ Where I wish to solve/optimise $R(h)$ (leaving $A(h) = const/h$) or $R(h)$ and $A(h)$ ...
2
votes
2answers
47 views

What is the answer to $\int x(t)dt$?

$\int x(t)dt$? I'm trying to solve a differential equation, but I've hit a strange brick wall that I never used to have a problem climbing over. This question is about mechanics & the equation ...
1
vote
0answers
24 views

How to evalute $\displaystyle\int_{2}^{x}\frac{t\ dt}{(\ln^{m} t)(\ln^{n} (t+2))}$

How to evaluate this integral? $$\int_{2}^{x}\frac{t\ dt}{(\ln^{m} t)(\ln^{n} (t+2))}$$ Where $m$ and $n$ are positive integers.
0
votes
1answer
167 views

Volume between paraboloid and plane

I need to find the volume of the finite region enclosed between the surface $$ y = 1 - x^2 - 4z^2 $$ and the plane $$y = 0$$ Here's what I've done: $$ \int\int ...
1
vote
1answer
40 views

Integral inequality with first two moments equal to $1$.

Let $f\in \mathcal{C}^0([0,1],\mathbb{R})$ such that $$ \int_0^1 f(x)\text{d}x = \int_0^1 xf(x)\text{d}x=1.$$ Show that $\int_0^1 f(x)^2 \ge 4$. I tried to use Cauchy-Schwartz inequality such that ...
7
votes
3answers
113 views

Why is $\displaystyle\int^{\infty}_{0}{(1-\cos x)\over{x^{2}}}dx = \frac\pi{2}$?

I have been having trouble understanding Fourier series and analysis in one of my classes. This is one problem from the text and we have to show that this is true. I have done other problems related ...
0
votes
1answer
38 views

Support of $L^p$ functions?

I noticed something strange. If we look at a function $f \in L^p$, then this is an equivalence class. Hypothetically: $\operatorname{supp}(f) = \overline{\{f\neq 0\}}$. But this is strange, as $f$ is ...
31
votes
1answer
835 views

Some users are mind bogglingly skilled at integration. How did they get there?

Looking through old problems, it is not difficult to see that some users are beyond incredible at computing integrals. It only took a couple seconds to dig up an example like this. Especially in a ...
1
vote
0answers
37 views

Convolving two functions

I'm trying to convolve two functions $f$ and $g$. $$f(x) = e^{-\frac{{(x-p_2)}^2}{2 q_2^2}}$$ $$g(x) = \left(i_1 e^{-\frac{(a-x)^2}{2 \sigma ^2}}+j_1 e^{-\frac{(b-x)^2}{2 \sigma ^2}}\right) \left(i_0 ...
0
votes
1answer
229 views

Moments of inertia of a torus

So, I can show that the moment of inertia of a torus about its axis of symmetry is $I_z = 4\pi^2\rho r^5\left[ \frac{3a}{8b} + \frac{a^3}{2b^3}\right]$ where $a$ is the distance from the axis to the ...
0
votes
0answers
197 views

Generate Borel Sigma Algebra

I want to show that the Borel Sigma-Algebra on $\mathbb{R}^n$ is generated by $ A:= \{(a_1,b_1] \times \cdots\times (a_n,b_n]; a_i,b_i \in \mathbb{R} \}$ as well as $ B:= \{(-\infty,c_1] ...
0
votes
3answers
90 views

Calculating a double integral

Calculate $$\int_{D}(x-2y)^2\sin(x+2y)\,dx\,dy$$ where $D$ is a triangle with vertices in $(0,0), (2\pi,0),(0,\pi)$. I've tried using the substitution $g(u,v)=(2\pi u, \pi v)$ to make it a BIT ...
1
vote
2answers
81 views

Show the function is integrable and find the integral - somewhat complex question

We are given $Q = [0,1]$x$[0,1]$ We are also given the function $f(x,y) = (\frac{1}{10})^n$ where $\frac{1}{2^{n+1}} < \max(x,y) \leq \frac{1}{2^n}, (n=0,1,2,...)$ and $f(0,0)=0$. Show that $f$ ...
2
votes
4answers
93 views

Evaluating the improper integral $\int_0^{\infty} \frac{\sin x}{x+x^2} \ dx$

Evaluating the improper integral $$\int_0^{\infty} \frac{\sin x}{x+x^2} \ dx$$ I'm trying to determine if the integral exists. I can't seem to deal with $$\lim_{a\to 0^+} \int_a^\infty \frac{\sin ...
0
votes
1answer
32 views

I need to show that this sequence is increasing and I'm almost there but I need help on last step.

Let $(1+\frac{1}{n})^n$ be a sequence and $f(x)=(1+\frac{1}{x})^x $ on $[1,inf)$. I need to show that f is non-decreasing by showing that $f'(x)\ge0$. So far I have: Let $g(x)=ln(f(x))$, where $ln$ ...
0
votes
2answers
34 views

Calculus power series

Hi could anyone help me to solve this. express the function $\int_x ^0 (\sin(t^2)\cdot \cos(t^2))$ as a power series. Because there is two trigo identies I do not know how to combine them to form a ...
0
votes
1answer
261 views

L1 convergence and Lp bounded implies Lq convergence

I have tried to solve this problem for almost a week and did not manage to, so I figured to ask it here: Let $(u_n)\to u$ in $L^1(0,1)$ strongly and let $\{u_n\}_{n\in\mathbb{N}}$ be bounded in ...
0
votes
1answer
36 views

Show that for $|f(z)| \leq C (|z| + 1)\log(|z| + 1)$, there is an $a$ such that $f(z) = az$

Let $f: \mathbb{C} \to \mathbb{C}$ be analytic and suppose a $C \geq 0$ exists such that \begin{align*} |f(z)| \leq C(|z| + 1) \log(|z| + 1) \end{align*} for all $z \in \mathbb{C}$, where $\log: ...
3
votes
2answers
111 views

a geometric interpretation of a line integral

Is there a geometric interpretation of the line integrals : $\int_{\gamma} f(x,y)\, dx$ $\int_{\gamma} f(x,y)\, dy$ (which should not be confused with $\int_{\gamma} f(x,y)\, ds$) where ...
0
votes
1answer
50 views

A question about improper integral

Could you please give me some hint how to solve this problem: Suppose f(x) continuous in $[0,\infty)$ and for each a,b>0 and c>b $ab \left|\int_0^1 f\left(ax+c \right) dx \right|<1$. Prove ...
2
votes
2answers
82 views

Computing $\int_{\gamma} {dz \over (z-3)(z)}$

Compute, using the Cauchy Integral Formula, $$ \int_{\gamma} {dz \over (z-3)(z)} $$ where $\gamma$ is the circle of radius $2$ centered at the origin, oriented counterclockwise. ...
1
vote
1answer
223 views

Line integral - should I parametrize the square?

I have the following $1-\text{form}$ defined: $$\omega = \displaystyle\frac{2xy}{(1-x^2)^2+y^2}\mathrm{dx}+\displaystyle\frac{1-x^2}{(1-x^2)^2+y^2}\mathrm{dy}$$ I'd like to find ...
4
votes
1answer
117 views

Show $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy =0 $

Need to prove $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy =0 $ and $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{y}{1+y^2}e^{-ay} dy =0 $ Can ...
1
vote
2answers
43 views

Triple integral problem involving definite integrals (and Taylor's formula possibly)

Any hint on how to approach this question? Show that $$\int_0^x\int_0^v\int_0^u f(t)dtdudv=\frac12\int_0^x(x-t)^2f(t)dt$$ I am completely clueless. I tried to convert it into the standard xyz form ...
20
votes
1answer
378 views

How to find $\int_0^\infty \prod_{k=1}^n \frac{\sin \frac{x}{2k-1}}{\frac{x}{2k-1}}\mathrm dx$

I am trying to calculate the integral $$ I_n=\int \limits_0^\infty \prod_{k=1}^n \frac{\sin \frac{x}{2k-1}}{\frac{x}{2k-1}}\mathrm dx. $$ (I have literature on this, if people want). Note, we can ...
3
votes
1answer
42 views

Properties of a Mehler's type integral

When computing the resolvent of the Laplace beltrami opetator on $S^n$ for even dimension, $n=2k$, I came across the following integral $$ ...
5
votes
1answer
117 views

Asymptotics of an oscillatory integral with a linear oscillator

I am interested in asymptotic results for $$ S(p) = \int_0^1 \frac{y \sqrt{1-y^2}}{(\varepsilon^2-1)y^2+1} \sin(py) dy, $$ i.e. a result that is valid as $p\rightarrow\infty$. The parameter ...
0
votes
3answers
60 views

Integrating the product of Poisson and exponential pdf

So I'll spare the background as to why, but I'm trying to integrate the following: $$\int_0^{\infty} \frac{e^{-(\lambda+\mu)t}(\lambda t)^n}{n!} dt$$ If you parameterize a Poisson w/ $\lambda$ and ...
0
votes
1answer
33 views

How to solve this definite integral from fourier series?

I am stuck with this integral: $$\int\limits_0^{2L} \sin \left( \frac{m \pi x}{L} \right) \sin \left( \frac{n \pi x}{L} \right)\, dx.$$ How to solve this integral? In general, can you refer me to ...
2
votes
1answer
41 views

Finding the flux integral?

Evaluate the flux integral $FdS$ where $F=\langle5y,2z,3x\rangle$ and $S$ is the part of the plane $6x+2y+z=12$ in the first octant oriented upward. This is how I solved it but the answer is ...
3
votes
0answers
40 views

Integration over time by having derivation

Assume we want to find the following integration: \begin{equation}\int_{t=0}^{\infty} p(t)dt\end{equation} where $p(0)=p$ and also $$\frac{dp(t)}{dt}=-p(t)(1-p(t))\mu$$. Is there any easy way to ...
1
vote
0answers
37 views

approximating a sphere

Suppose that $R$ is a simple connected region in $\mathbb{R}^3$, enclosing a volume $V$. I am looking at ways to approximate $V$ using spheroidal volume elements. The traditional approach is to use ...
1
vote
1answer
31 views

How to find f(x+a)?

Let f be a function defined by $$f(x)=\int_1^x\frac{e^t}{t}dt,x>0$$ Then we have to find the value of $$\int_1^x\frac{e^t}{t+a}dt$$ Here's what I did: ...