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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3
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1answer
45 views

Different results for $\int {2}/(3x+1)\; dx$

I was reviewing some old calculus notes, when I noticed an odd example of integral of rational functions: $$ \int \frac{2}{3x+1}\; dx $$ And depending on how it is resolved, it gets diferent values. ...
0
votes
1answer
37 views

When is $\lim_{\delta x_0\to0} \int_{x_0}^{x_0+\delta x_0}f(x)dx=0$?

$\int_{x_0}^{x_0} f(x) = 0$ if $f(x_0)$ is defined. Then, for functions that are "nice enough", we should have $$ \begin{align} \lim_{\delta x_0\to0} \int_{x_0}^{x_0+\delta x_0}f(x)dx=0 \tag1 ...
0
votes
0answers
36 views

How to do contour integration? [duplicate]

I've searched a lot throughout the web, but havent found anything yet, so I am posting my own question. I am very interested in complex analysis, hopefully someone can help me out here. Suppose we ...
0
votes
2answers
48 views

Center of mass Double Integral?

Can you help me with this problem? Find the center of mass of a lamina whose region $R$ is given by the inequality: $$|x|+|y|\le 1,$$ and the density in the point $(x,y)$ is : ...
2
votes
4answers
76 views

This improper integral $\int_{1}^{+\infty}x\sin{x}\sin{x^4}dx$ is absolutely convergent?

Discuss the improper integral $$\int_{1}^{+\infty}x\sin{x}\sin{x^4}dx$$ absolute convergence? My idea: since $$\sin{x}=x-\dfrac{x^3}{6}+\cdots$$ $$\sin{x^4}=x^4-\dfrac{x^{12}}{6}+\cdots$$ so ...
0
votes
1answer
39 views

Integration and function composition

I would like to know which are the conditions on the continuous functions $h$ and $g$ for the following equality to be true: $$ \int_{X} (h \circ g) (x)dx = \int_{W} h(w)dw, $$ where $X$ is an ...
2
votes
1answer
55 views

Conservative vector field, potential function and work done

For (i), is that I have to show $curl F = 0$ ? For (ii) and (iii), what should I do in order to find the potential function and work done? Also, is the answer $4$ for (iii)?
1
vote
1answer
39 views

Multiple integrals: Double integrals

For this question, how to evaluate the integral by changing the order of integration? Also, how to sketch the region of integration? I really get stuck.
0
votes
0answers
56 views

How to bound this triple integral?

Find the volume of the solid bounded by $x^2+y^2 = \frac{1}{2}$, $x^2+y^2 = 1$, $x^2+y^2-2y = 0$, $x \ge 0$, $y \ge 0$, $z = e^{x^3} + x*(x^2+y^2)$, $z = e^{x^3}$ By setting $z = z$, we found the ...
2
votes
1answer
46 views

Double Integral Between Two Circles

What is the area confined by these two circles? $y^2=1-(x-1)^2$ $y^2=1-x^2$ I've set up the integral: $\int_{\pi/3}^{2\pi/3}\int_{2\cos\theta}^1r\,\mathrm{d}r\mathrm{d}\theta$ Unfortunately, my ...
1
vote
3answers
71 views

Calculate center of mass multiple integrals

Can you help me with this problem? Find the center of mass of a lamina whose region R is given by the inequality: and the density in the point (x,y) is : The region r is this one: Is this the ...
2
votes
1answer
51 views

why $\int_{-\infty}^{\infty} \left\vert e^{-ax} \right\vert^{2} dx = \int_{0}^{\infty} e^{-2ax}dx$

The calculus text book says $\int_{-\infty}^{\infty} \left\vert e^{-ax} \right\vert^{2} dx = \int_{0}^{\infty} e^{-2ax}dx$ but does not explain how this happened, and I am not able to figure it out. ...
0
votes
1answer
28 views

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

Not sure where I'm going wrong on this one. $$\int{ye^{-(x+1)y}}\:dy$$ $$u = y \qquad du = dy$$ $$dv = e^{-(x+1)y} \qquad v = -\frac{e^{-(x+1)y}}{x + 1}$$ $$-\frac{ye^{-(x+1)y}}{x + 1} \times ...
0
votes
2answers
37 views

Double integral with polar?

I have the following integral : $$\iint\limits_R \operatorname e^{-\frac{x^2+y^2}{2}} \operatorname d\!y \operatorname d\!x $$ Where R is: $$R=\{(x,y):x^2+y^2 \leq 1\}$$ I think I should convert to ...
3
votes
1answer
30 views

A convenient variable change for solving an integral

I'm trying to compute the following integral: $$ \int_0^1\int_{x^2}^x (x^2+y^2)^{\frac{-1}{2}} \, dy dx$$ I already noted that this is the region between the parabola $y=x^2$ and the line $y=x$, ...
1
vote
2answers
44 views

Solve double integral

$$ \int_0^2 \int_0^{4-x^2} \frac{xe^{2y}}{4-y} \, dy\, dx $$ I'm stuck with this problem. I think I should change it so I integrate with respect to $dx \, dy$ but I'm not sure. Any help? Thanks
0
votes
1answer
31 views

Solve the double integral

I am calculating: $$ \int\int_R (2ax-x^2-y^2)^{\frac{1}{2}} \, dA$$ Where $R$ is the region determined by the inside of $x^2+y^2-2ax=0$ So far, I tried using polar coordinates, wich turns the ...
0
votes
0answers
30 views

Calculate center of mass [duplicate]

Can you help me with this problem? Find the center of mass of a lamina whose region R is given by the inequality: and the density in the point (x,y) is : Any help? Thanks
0
votes
1answer
16 views

Graphing in Polar Coordinates

I´m currently using polar coordinates to calculate some double and triple integrals. However, I have an small doubt; when you are want to express, lets say, a circle of radius $a$ centered in $(a,0)$ ...
0
votes
2answers
30 views

Help understanding the limits of this integration problem

Find $P(X < 2Y)$ if $f_{X, Y}(x, y) = x + y$ for $X$ and $Y$ each defined over the unit interval. Now if I understand the problem correctly, that should mean a triangle with the vertices $(0, 0), ...
0
votes
2answers
85 views

Definite integral of $\sin(x^3)$

I am having trouble integrating $\sin\left(\, x^{3}\,\right)$ from $x = -10$ to $x = 10$. I have tried $u$ substitution but is doesn't seem to work out. By the way the question is $\sin\left(\, ...
1
vote
1answer
38 views

Need help bounding triple integral

I'm studying for an exam and I struggle with finding the bounds for triple integrals. Specifically, I find it difficult when I can not draw a picture of the surface (which I may not have time to do on ...
6
votes
1answer
154 views

Banach Spaces: Uniform Integral vs. Riemann Integral

Problem Given a finite measure space $\Omega$ and a Banach space $E$. One has strict inclusion: ...
-1
votes
2answers
57 views

Non-homogeneous Differential Equation (Stuck at integration part)

I need a hand for solving the integration part of the differential equation $y''+4y=x^2sin2x$ . $(D-2i)(D+2i)y=x^2sin2x$ , $t= \dfrac{x{^2}sin2x}{D+2i}$ $t'+2it=x^2sin2x$, $t=uv$ $v=e^{-2ix}$ ...
2
votes
2answers
76 views

Inequality $\left| \int_{-\pi}^{\pi} \varphi(t) \sin(n t) \operatorname{d} t \right| \leq \frac{2 \pi}{n} \max_{x \in [-\pi, \pi]} |\varphi(x)|$

I have a monotonic increasing function $\varphi$ which is continious and not negative with domain $[-\pi,\pi]$. I want to show that for all $n \in \mathbb N$: $$ \left| \int_{-\pi}^{\pi} \varphi(t) ...
2
votes
1answer
66 views

Determine how large the number a has to be?

This is what i've done so far: i converted to limit notation lim as t goes to infinity of integral from a to t of 1/t^2+1 dt lim as to goes to infinity [arctan(t)] from a to t (lim as t goes to ...
0
votes
0answers
22 views

question about simplifying the integral(appeared in the proof of interpolation theorem of BMO and L^p)

I do not know how to calculate the following integral, which I saw when reading a short note about interpolation result about BMO spaces. $r\displaystyle\int_{2^{n/p}}^\infty \lambda^{r-1} c_1 ...
3
votes
1answer
80 views

$\displaystyle\int e^{\frac{x^3}{3}} \mathrm{d}x$

While solving $y'=x^2-e^y$ I'm stuck on the last step that requires to evaluate this integral. $$\displaystyle\int e^{\frac{x^3}{3}} \mathrm{d}x$$ I don't know how to approach it. I know that it ...
0
votes
1answer
41 views

Integral of $\int_0^{\infty} x^{4n+3} e^{-x} \sin x dx$.

Can some one help me with the integral $$\int_0^{\infty} x^{4n+3} e^{-x} \sin x dx$$ According to my exercise I should be able to get $0$. Please help me .
11
votes
3answers
408 views

Closed form of $\int_{0}^{\infty} \frac{\tanh(x)\,\tanh(2x)}{x^2}\;dx$

I have homework to evaluate this integral $$I=\int_{0}^{\infty} \frac{\tanh(x)\,\tanh(2x)}{x^2}\;dx$$ Here is what I have done so far. I tried integration by parts using $u=\tanh(x)\,\tanh(2x)$ and ...
0
votes
1answer
40 views

search for a theorem related to $\sum\limits_{n=1}^{\infty}n^2\exp(-n^2)<\int_{1}^{\infty}\nu^2\exp(-\nu^2)d\nu$

I need to use the following inequality: $$\sum_{n=1}^{\infty}n^2\exp(-n^2)<\int_{1}^{\infty}\nu^2\exp(-\nu^2)d\nu\tag{1}$$ what is the name of such a theorem?
0
votes
1answer
63 views

What does it mean for an integral to “vanish”?

I had a question; What does it mean for an integral to vanish in complex analysis? There is supposedly something, which says if the integral "vanishes," the sum of the residues is 0. But what does ...
2
votes
1answer
51 views

Integral inequality with sines

I am trying to show that there exists some constant $\mathcal{C}>0$ such that: $$\mathcal{C}\leq \int_0^1 |\sin (2\pi n x)-\sin (2\pi m x)|\;dx$$ For all distinct $m,n\in\mathbb{N}$. The constant ...
8
votes
1answer
161 views

Alternative ways to evaluate $\displaystyle \int^1_0 \frac{\text{Li}_2(x)^3}{x}\,dx$

In the following link here I found the integral & the evaluation of $$\displaystyle \int^1_0 \frac{\text{Li}_2(x)^3}{x}\,dx$$ I'll also include a simpler version together with the question: is ...
2
votes
1answer
121 views

Solving ODE $F(t)=A(t)F'(t) $

How to solve $F(t)=A(t)F'(t) ,F(0)= I\tag 1$ All are $3 \times 3$ matrices except variable t A(t) is given and has determinant $0$. $A(t)=(I-tC_1)^{-1}t^3C_2 \tag 2$ I is a constant unit ...
1
vote
1answer
25 views

Change of variable integral

Consider the following integral \begin{equation} \int_{\Omega} f(y,My-z_2)\, g(z_1,z_2,y) ~ dz_1\, dz_2\, dy \end{equation} $f(y,My-z_2) = 1$ (a constant function for each value of $y$ and $z_2$) ...
1
vote
6answers
88 views

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

I want to solve $$\int \frac{1}{\sqrt{x^2 - c}} dx\quad\quad\text{c is a constant}$$ How do I do this? It looks like it is close to being an $\operatorname{arcsin}$? I would have thought I could ...
2
votes
1answer
59 views

Find the density of the random variable with characteristic function $\varphi(t)=(1-|t|)^+$.

Suppose a $X$ is a random variable, I am asked to find the density of the random variable with characteristic function $\varphi(t)=(1-|t|)^+$. I am trying to use the inversion formula for the ...
0
votes
2answers
38 views

How to do this Integration (in Orthonormal Family for Continuous Functions)

Here is a common example in the discussion on orthonormal family. Let $\mathcal L = C[a, b]$. For $k \in \mathbb Z$, let $e_k \in \mathcal L$ be defined by $$e_k(\xi) := \frac{1}{\sqrt{b-a}} ...
0
votes
1answer
62 views

An integral involves Gamma function

Thanks for your attention, I meet an integral involves Gamma function and exponential function as follows:$$\int_a^\infty {{x^\alpha }} {e^{cx}}\Gamma \left( {s,bx} \right)dx$$ where $a > 0,s ...
1
vote
3answers
78 views

Does $\sum \frac{(n+4^n)}{n+6^n}$ converge or diverge?

The Question Does $\sum \frac{(n+4^n)}{n+6^n}$ converge or diverge? Please note I have no knowledge of Alternating Series, Ratio and Root tests, Power Series, or Taylor and McLaurin Series. My Work ...
1
vote
1answer
38 views

Curve parameterization trick

So, I was given this really nasty problem to solve Suppose C is parametrized by $\mathbf{g}(t) = \left[\begin{array}{c}e^{t^{3}\cos\!\left(2\pi t^{25}\right)}\cr t^{6}+3t^{3}+3\cr ...
3
votes
2answers
111 views

What is the correct definition of Area?

How is the area of a rectangle: length $\times$ breadth? We know that other areas can be derived from it. Also, the area under curves uses the area of rectangles as a basis.
1
vote
0answers
95 views

Triple integral-pyramid

Let the pyramid with vertices $A(0,0,0), B(0,0,1), C(0,1,0), D(1,1,0)$. I need to find the equations of the four planes bounding the pyramid then I have to set up an integral for the volume in three ...
6
votes
3answers
118 views

How to evaluate this improper integral?

I got stuck when evaluating these two improper integrals:$$ \int_a^b\frac{dx}{\sqrt{(b-x)(x-a)}} $$ and$$ \int_0^1\frac{dx}{\sqrt{x-x^3}} $$ How to evaluate them? Thank you!
1
vote
2answers
61 views

Mean of a Cauchy Distribution

Why is the mean of a Cauchy distribution undefined? Surely, it should be $0$ by symmetry? $$\int_{-\infty}^{\infty} {\frac{x}{\pi (1+x^2)}} dx =0?$$
1
vote
2answers
45 views

Is $f\left(t\right)=\frac{1}{t^2+1}$ of exponential order?

I'm learning Laplace Transforms and one of the questions I'm working on is the following: $$\text{Is}\:\:f\left(t\right)=\frac{1}{t^2+1}\:\:\:\text{of exponential order?}$$ If so or if not, how do I ...
2
votes
3answers
99 views

limit $\lim_{n\rightarrow \infty}\left(\int_0^1f^n(x)dx\right)^{\frac{1}{n}}=M$

For a continuous positive funciton $f(x)$ on $[0,1]$, with maximum value $M$, show that $$\lim_{n\rightarrow \infty}\left(\int_0^1f^n(x)dx\right)^{\frac{1}{n}}=M$$
0
votes
1answer
53 views

Shifted Fourier transform

Please can some one help and give me a direction to evaluate the following shifted Fourier transform: \begin{alignat}{2} s(x_c) =&\frac{1}{\Delta x_0} \int_{x_c-\Delta x_0}^{x_c+\Delta ...
0
votes
3answers
30 views

integration help with area

Find the area of the region bounded by the curves of $y=2x^2-3x+5$ and $y=x+11$. So far I have done this: $2x^2-3x+5 = x+11$ $2x^2-4x-6=0$ $2(x^2-2x-3)=0$ $(x-3)(x+1)=0$ $X=3, X=-1$ What do I ...