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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11 views

Evaluating $\int\exp\left(\frac{x-y}{x+y}\right)\mathrm dx$

I'm stuck on this simple double integral: $$\iint_T\exp\left(\frac{x-y}{x+y}\right)\mathrm dx\mathrm dy = \frac14\left(e-\frac1e\right),$$ with $T = \{(x,y) \in \mathbb R^2 \colon x > 0, y > 0, ...
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0answers
16 views

Integral $\frac{sin(x)}{x}$ finite domain

I have seen a question asking to find the value of $\int_{-100}^{100} \frac{\sin{x}}{x} dx$. I have to confess that I didn't think this was possible. If I expand the $\sin$ using Taylor series, ...
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0answers
14 views

Change of variables from unit unit ball to another ball for integration

What is the general formula for changing coordinates for integration from the unit ball to another ball? For example, if I wanted to change from integrating $f(x-r)$ over $B(0,1)$, the open ball about ...
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3answers
67 views

Integrate $\int{\frac{1}{(x+1)(x+2)^2(x+3)^3}dx}$

How to integrate this $$\int{\frac{1}{(x+1)(x+2)^2(x+3)^3}dx}$$ I tried to use that $$\int{\frac{1}{(x+1)(x+2)^2(x+3)^3}dx} = P_{1}(x)/Q_{1}(x) + \int{P_{2}(x)/Q_{2}(x)dx}$$ where ...
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0answers
29 views

Indefinite integration and definite integration calculation

Given the functions: $$\begin{cases} A(x)= \dfrac {x\ln(1+\ln(x))}{1+x^{4/3}}\\[2ex] B(x)=A(2x-1)-A(2x) \end{cases}$$ for $b>1$ find out: $$\int_{b}^\infty A(x) dx - \frac12\int_{2b-1}^{2b} B(x) ...
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1answer
15 views

Trapezoid rule for finding coefficient

If we know that $\int_{a}^b t(x)=h \sum_{k=1}^2 dk * t(a+kh)+O(h^m)$ where $h=\frac{b-a}{3}$, how do we find the coefficient d1, d2 and m in the equation? Answer says that d1=3/2, d2=3/2, m=3 I ...
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2answers
86 views

Anti-derivative of continuous function $\frac{1}{2+\sin x}$

I use tangent half-angle substitution to calculate this indefinite integral: $$ \int \frac{1}{2+\sin x}\,dx = \frac{2}{\sqrt{3}}\tan^{-1}\frac{2\tan \frac{x}{2}+1}{\sqrt{3}}+\text{constant}. $$ ...
0
votes
2answers
27 views

Integral of a measurable function

I do not know what should i keep as title for this question... Question goes like this.. Let $f:\mathbb{R}\rightarrow [0,\infty)$ be a measurable function. If $\int_{-\infty}^{\infty}f(x)dx=1$ prove ...
2
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0answers
40 views

difficult integral $\int_0^{\pi/2}\frac{x^2({1+\tan^2 x})^2}{\sqrt{\tan x}({1-\tan x})}\sin{4x}dx$

This is a complicated integral, the numerical value appears to me correct.Therefore how to prove this result?$$I=\int_0^{\pi/2}\frac{x^2({1+\tan^2 x})^2}{\sqrt{\tan x}({1-\tan ...
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0answers
61 views

Solving an indefinite integral problem [on hold]

The given problem is $$ \int \frac{2^{\sin x}}{2^{\sin x} + 2^{\cos x}} dx$$ please help me solving this indefinite integral problem...thank you very much, actually I have solved an definite integral ...
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0answers
25 views

Proving an inequality involving integrals?

I am trying to prove that $$[\sum_{i=1}^{n}(\ln t_i)^2 t_i^\alpha+A^{\prime \prime}(\alpha)][\sum_{i=1}^{n}t_i^\alpha+A(\alpha)]\ge[\sum_{i=1}^{n}(\ln t_i) t_i^\alpha+A^{\prime}(\alpha)]^2$$ where ...
0
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0answers
44 views

how to evalute this equality

I want to prove this equality $$ \frac{1}{2\pi}\frac{(x-y)\cdot y}{(x_1-y_1)^2+(x_2-y_2)^2}= \frac{ab}{4\pi}\frac{1}{a^2\sin^2(\alpha+\beta)+b^2\cos^2(\alpha+\beta)}.\tag{1}$$ where ...
0
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1answer
19 views

Integral, partial fractions, need explanation for how to get from one step to another.

Can someone explain how they go from the red step to the blue one?
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1answer
25 views

How to Proceed in Solving this Equation

Let $f: [0,\infty)\to \mathbb{R}$ a non-decreasing function. Then show this inequality holds for all $x,y,z$ such that $0\le x<y<z$. \begin{align*} & (z-x)\int_{y}^{z}f(u)\,\mathrm{du}\ge ...
1
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1answer
26 views

Proving a reduction formula. $\cos^n (2x)$

Establish a reduction formula for $$\int \cos^n (2x)dx$$ My attempt, Let $I_n=\int \cos^n 2x dx$ $=\int \cos^{n-1}2x (\cos 2x dx)$ Let$$u=\cos^{n-1}2x$$ $$du=-2(n-1)\cos^{n-2}2x (\sin 2x)dx$$ ...
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0answers
26 views

Bounding an integral

I'm trying to show that the following integral ( a solution for the non-homogeneous transport equation ) has this bound: $$ \begin{equation*} \left\|{ \int_{0}^{t} f(x+b(w-t),w) dw ...
1
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2answers
39 views

Sign of the error in Simpson's rule

Let $f : [a,b] \to \mathbb{R}$ be a $C^\infty$ function. The Riemann integral $I = \int_a^b f(x)\,dx$ can be approximated by using Simpson's rule: $$I \approx S = \frac{b-a}{6} \left[ f(a) + 4 ...
3
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0answers
42 views

An alternative way to determine when $\int_{0}^{\infty} \cos(\alpha x) \prod_{m=1}^{n} J_{0}(\beta_{m} x) \, dx =0$

Let $J_{0}(z)$ be the Bessel function of the first kind of order zero, and assume that $\alpha$ and $\beta_{m}$ are positive real parameters. When $|z|$ is large in magnitude and $-\pi < \arg(z) ...
1
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1answer
66 views

Integrate $\int{ \left( \frac{1-x}{1+x} \right)^\frac{3}{2}dx}$

Integrate $$\int{ \left(\frac{1-x}{1+x} \right)^\frac{3}{2}dx}$$ I guess that there is sub $x = \cos t$ so integral gets to $$\int{ \left(\tan \frac{t}{2} \right)^3 d\cos t}$$ then I used that $\sin t ...
0
votes
1answer
25 views

Surface are of a curve $y=\sin \left(\frac{\pi x}{6} \right)$ rotated about the $x$ axis.

I'm doing a problem involving finding the surface area of the curve for $y=\sin \left(\frac{\pi x}{6} \right)$, rotated about the $x$ axis, for $[0 < x < 6]$. I got as far as $\frac{72}{\pi} ...
0
votes
1answer
22 views

Problem with convolution, insecure

$$f(t)= t^2\cdot u(t),\quad g(t)=t^4\cdot u(t)$$ I know that I need to use convolution theorem to solve this problem, but I really don't know what to do with step functions. Do I need to include ...
0
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0answers
29 views

Evaluating this surface integral

I want to evaluate the following surface integral $$\int_S d S \, \, \exp\left(\frac{\Gamma^2 (x^2 \sigma_x^2 + y^2 \sigma_y^2 + z^2 \sigma_z^2) - 2 c R \Gamma (x x_0 + y y_0 + z z_0)}{2 c^2 ...
3
votes
4answers
72 views

Clever way of calculating the integral $ \int \frac{dt}{t^2\sqrt{t-2} } $

$$ \int \frac{\text{d}t}{t^2\sqrt{t-2} } $$ I know it can be calculated using somewhat complicated substitutions, but is there possibly some clever way of solving that type of integral? I don't ...
1
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0answers
17 views

Integral of least squares and general rules of integration to solve the integral.

My calculus is very rusty and I am interested to know if the following is solvable: $$ \int_0^{\pi}( \log( \frac {(x_0 + e^{-i\omega})(x_0 + e^{i\omega})(x_1 + ...
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2answers
55 views

How would you calculate $(200\int_0^\infty e^{-0.8t}-e^{-1.8t}\,dt)/(250\int_0^\infty e^{-0.8t} \,dt)$?

$$\frac{200\int_0^\infty e^{-0.8t}-e^{-1.8t} \, dt}{250\int_0^\infty e^{-0.8t} \, dt}$$ I am confused as to how you would integrate the e's from zero to infinity. What steps would you take? By the ...
1
vote
0answers
31 views

reduction formula for $\int \tan^n (2x)dx$

Establish a reduction formula for $$\int \tan^n (2x)dx$$ My attempt, Let $I_{n}=\int \tan^n (2x)dx$ $=\int \tan^2 (2x) \tan^{n-2} (2x)dx$ $=\int (\sec^2 (2x)-1)\tan^{n-2}(2x)dx$ $=\int ...
1
vote
0answers
11 views

Prove that the condition $x(\tau)>\xi$ of a divergent integral implies that $x(t)>\xi$

Let $ E, J \subset \mathbb R$ be open intervals and let functions $h:J \to \mathbb R$ and $g: E \to \mathbb R$ be continuous. let $\xi \in E$ and assume that $g(\xi)=0$. Define $f:J \times E \to ...
2
votes
3answers
71 views

Calculate $\int_0^1 \ \int_0^1 \ x \sin \lvert x^2-y^2 \lvert \; dx \; dy$

$$\int_0^1 \ \int_0^1 \ x \ \sin \lvert x^2-y^2 \lvert dx \ dy $$ $$\int_0^1 \frac{1}{2} \Big[ \sin \lvert x^2-y^2 \lvert \Big]_0^1 \ dy= \int_0^1 \frac{1}{2} \Big( \sin \lvert 1-y^2 \lvert - ...
4
votes
6answers
120 views

integrate $\int \frac{dx}{x\sqrt{1-x}}$

$$\int \frac{dx}{x\sqrt{1-x}}$$ $$\int \frac{dx}{x\sqrt{1-x}}$$ $u=1-x$ $du=-dx$ $$-\int \frac{du}{(1-u)\sqrt{u}}$$ $a(1-u)+b\sqrt{u}=1\Rightarrow a-au+b\sqrt{u}=1$ $a=1\Rightarrow ...
2
votes
2answers
57 views

How to solve without involving hyperbolic function.

How to solve this integral without involving hyperbolic functions? $$\int \frac{1}{4-5\sin^2 x}dx$$ The answer is $\frac{1}{4}(\ln (\sin x+2 \cos x)-\ln(2\cos x-\sin x))+c$
2
votes
1answer
38 views

Imaginary number and absolute value integral - Fourier transform

I came across this integral problem: $$\hat f(\xi)=\int_{-\infty}^{+\infty} e^{-|x|+xi\xi}dx$$ Now I know how to integrate simple absolute value functions like: $\int_{-2}^{4}|x-2| dx$, we just ...
-1
votes
1answer
36 views

Density of $L^\infty(\Omega)h$ in $L^p(\Omega)$ where $h \in L^p(\Omega)$

Let $(\Omega,\mu)$ be a finite measure space. Suppose $1\leq p <\infty$. Let $h$ be an element of $L^p(\Omega)$ with $h >0$ a.e.. How show that the subspace $L^\infty(\Omega)h=\{ f h\ :\ f\in ...
0
votes
2answers
55 views

Indefinite trignometric integral

I tried $u$-substitution and $uv$-substitution, can't seem to figure this out... any help would be appreciated! Question: $$\int\frac{x}{\cos(x)}\,dx$$ Thanks!!!
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3answers
27 views

How to find differentiation and integration of curves in general?

Graph of function $f(x)$ How do I go about finding integration and differentiation of curves like these which yield other curves?
0
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1answer
45 views

Finding Value of C to Maximize Area

f(x)=$xe^{-\sqrt x}$ Find the value of c, such that the area bounded between the graph, the x-axis, x=c, and x=c+1 is maximized. Find the maximum area. I don't know where to start with this one. I ...
0
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3answers
26 views

How to account for solids of revolution around vertical lines to the right of the x axis?

I'm trying to find the volume of a solid created by rotating the region enclosed between $x=y^2$ and $x=1$ around the line $x=8$. Noting that the intersections of the functions occur at $(0,0)$ and ...
0
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1answer
16 views

Vector integral in $N$ dimensions.

In $N$ dimensions I want to do an integral of the flux through an $N-1$ dimensional surface. The usual vector calculus integration theorems help by allowing integration around the perimeter of the ...
0
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1answer
25 views

Trouble getting between steps when solving integral

I've having a lot of trouble trying to figure out how they're getting from the step in blue to the one in red. Can some one please explain that?
0
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0answers
19 views

Estimate integration

Suppose $n<<k$ and $n<i<k$. How can I estimate this integral: $\int_{1}^{k-i} x^{-0.4}*(x+i-1)^{-0.4}dx$. I would like to get the result in the form of O(f(k,i)). Since the integrand is ...
2
votes
1answer
46 views

Finding Fourier series constant and integral

I have been studying Griffith's Intro to Electrodynamics. I am studying differential equations and Fourier series. I am studying the problem discussed here: Why is this allowed? ("Fourier's ...
2
votes
1answer
22 views

Volume of region in the first octant bounded by coordinate planes and a parabolic cylinder?

Find the volume of the solid region in the first octant bounded by the coordinate planes, the plane $y + z = 2$ and the parabolic cylinder $x = 4 - y^2$. I have a final answer, I would just like to ...
1
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2answers
69 views

Integrate $\int \frac{x^7}{(1-x^4)^2}dx$

$$\int \frac{x^7}{(1-x^4)^2}dx$$ I have tried to simplify the expression, to use U substation, any idea where to start from?
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0answers
31 views

Is there a close form expression for the integral $ \int_a^b |x-c|^n e^{-x^2/2} $

Is there a close form expression for the integral \begin{align} \int_a^b |x-c|^n e^{-x^2/2} dx \end{align} by close form I mean it can be in terms of well know functions such as $Q$-function, ...
0
votes
2answers
65 views

How do I evaluate a series? [on hold]

In this specific example, I don't understand the steps of evaluating this series: \begin{align} &\frac{12}{n}\left(\left[\sum_{i=1}^n-7\right]+\sum_{i=1}^n\left[\frac{-12}{n}\cdot ...
1
vote
2answers
49 views

Arc length of Archimedes Spiral $ r = \theta $ from $ 0 \le \theta \le 2\pi$

The equation of the Archimedes spiral is given by $$r = \theta$$ The formula for calculating the Arc Length is given by $$L = \int^b_a\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}d\theta$$ The ...
2
votes
0answers
39 views

Asymptotic behaviour of an integral depending on a parameter

I am trying to compute the asymptotics on $t$ of the following integral: \begin{equation} I(t)=\int_{\mathbb{R}^{n}}e^{-|\lambda|^{2}/2t}\prod_{i<j}\left( e^{\lambda_{j}/t}-e^{\lambda_{i}/t} ...
1
vote
7answers
102 views

Integrate $\int \frac{x\cos x}{\sin^2x}dx$

$$\int \frac{x\cos x}{\sin^2x}dx$$ $$\int \frac{x\cos x}{\sin^2x}dx=\int \frac{x\cos x}{1-\cos^2x}dx=\int \frac{x\cos x}{(1-\cos x)(1+\cos x)}dx$$ How can I find the two fractions? if there are ...
3
votes
0answers
51 views

How to calculate the area of the visible parts of a 3D PieChart?

I have created a 3D Pie Chart whose major feat (among the others) is to be rotated: I did it to demonstrate how the visual perception of data in a Pie Chart can be distorted depending on the ...
-1
votes
1answer
17 views

how to prove if $f$ is integrable and $f'$ too then the limit of $f$ is zero when $x$ go to infinity? [on hold]

If $f$ is a real function on $\mathbb R$ and we have $\int_1^\infty |f(x)|dx < \infty$ and $\int_1^\infty |f'(x)|dx < \infty$ then $\lim_{x\to\infty}f(x)=0$ ?
1
vote
0answers
42 views

Compute $\frac{d}{dt}\int_0^t e^{x(s)}ds$, where $x$ is a standard Brownian motion.

How to compute the following differentiation? Is there a general rule that can be applied? $$\frac{d}{dt}\int_0^t e^{x(s)}ds$$ in the case of $x=W$ where $W$ is a standard brownian motion, is there ...