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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2answers
67 views

Variation of Jensen-Inequality

I just read a variation of Jensen's Inequality which states: If $f: \mathbb{R} \rightarrow \mathbb{R} $ is a convex function, $ \phi \in \mathcal{L}^1(\mathbb{R}^n)$ with $ \phi \geq 0$ and $ \int ...
1
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1answer
52 views

Integral of power mean function

I wonder, is there a closed-form solution to integral of type $$\int \left(\frac{a^x+b^x}{2}\right)^{1/x}dx,$$ for $a\ne b$ (both positive real numbers). If not, what about a special case when $a=1$, ...
1
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2answers
59 views

Evaluating $\frac{d}{dx}\int_1^{x^2}\sqrt{y^2+3}dy $

How to Evaluate $$\frac{d}{dx}\int_1^{x^2}\sqrt{y^2+3}dy $$On doing differentiation of integration we obtain the same function but here limit of integration is also to be applied .How can we do it ...
5
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1answer
84 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?
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2answers
27 views

Evaluating $\frac{d}{dx} \int_{1}^{3x} \left(5\sin (t)-7e^{4t}+1\right)\,\mathrm dt$

$$\dfrac{d}{dx} \int_{1}^{3x} \left(5\sin (t)-7e^{4t}+1\right)\,\mathrm dt$$ The answer I come up with is: $5\sin(3x)(3)-7e^{4(3x)}(3)$, however this was not on the answer choice. What is the ...
2
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1answer
55 views

Evaluating derivative of $\int^{3x}_{2x} \sin(t^3 + 1) \,\mathrm dt$

Maybe I'm not very good at my trig rules but I'm having a tough time finding derivative of $$\int^{3x}_{2x} \sin(t^3 + 1) \,\mathrm dt$$ I believe that $u = t^3 + 1$ and $du = 3t^2$, but I'm not ...
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2answers
43 views

Producer Surplus

If a supply curve is modeled by the equation $$p = 600 + 0.1q ^{1.5}$$ find the producer surplus when the selling price is $\$700$. I did this problem multiple ways and still can't figure out what ...
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2answers
74 views

Prove that if $f$ is integrable on $[0,1]$, then $\lim_{n→∞}\int_{0}^{1} x^{n}f(x)dx = 0$.

Prove that if $f$ is integrable on $[0,1]$, then $\lim_{n→∞}\int_{0}^{1} x^{n}f(x)dx = 0$. Attempt: this exercise has two parts. I did part a) already. From part a) we know that $g_{n} \geq 0$ is ...
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1answer
33 views

Find the Moment of Inertia (Need Help With a Single Step)

Find the moment of inertia around the z axis of the region bounded by x=0, y=0, z=0, x+y+z = 1. My attempt at a solution: I believe that this should require a triple integral of the form ...
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1answer
22 views

Volume by rotation: Cyndrilical shells and radii

For a region R bounded above by the curve $y = e^{-x^2}$, below by the curve $y = x^2 - 1$, on the left by the curve $x = -1$, and on the right by $x = 1$, that is rotated around the vertical line ...
1
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2answers
57 views

Confusing property of definite integrals

Consider the proof of the property $$\int_{a}^{b}f(x)dx=\int_{a}^{b}f(a+b-x)dx$$ Solution :- Let $t = a + b - x$ then, $$\int_{a}^{b}f(x)dx=\int_{a}^{b}f(a+b-t) dt$$ The next step is ...
1
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1answer
50 views

Evaluating $\frac{d}{dx}\int_{7-2x}^3 \frac{u^3}{1+u^2}du$

Evaluate following expression by using The Fundamental Theorem of Calculus $$\frac{d}{dx}\int_{7-2x}^3 \frac{u^3}{1+u^2}du$$
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1answer
22 views

Identity involving the relation Normal Distribution and Other arbritary Distribution

Let $X$ be a continuous random variable taking value at $\mathbb R$ with distribution $F_\theta$ and density $f_\theta.$ Define a function $\psi_\theta:\mathbb R\mapsto \mathbb R$ by $$ ...
2
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1answer
49 views

Integral of a gaussian over a slice of the plane

I need to evaluate the following $n$ real integrals: $$\int_{\frac{\pi}{2}-\frac{\pi}{n}}^{\frac{\pi}{2}+\frac{\pi}{n}}\int_0^\infty\frac{1}{\pi \sigma^2}e^{-\frac{|re^{i\theta}-i|^2}{\sigma^2}} \ dr ...
12
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1answer
201 views

A fractional part integral giving $\frac{F_{n-1}}{F_n}-\frac{(-1)^n}{F_n^2}\ln\left(\!\frac{F_{n+2}-F_n\gamma}{F_{n+1}-F_n\gamma}\right)$

I've been asked to elaborate on the following evaluation: $$ \begin{align}\\ \displaystyle {\large\int_0^{1}} \!\cfrac 1 {1 + \cfrac 1 {1 + \cfrac 1 {\ddots + \cfrac 1 { 1 + \psi ...
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1answer
37 views

For every periodic continuous function $f$, the function $s\to \int_a^b f(x/s)\, dx $ is continuous

Let $f: \mathbb R \to \mathbb R$ be a continuous function such that $f(x+1)=f(x)$ for all $x\in \mathbb R$. Fix $a$ and $b$ such that $a<b$, and define a function $g: \mathbb R \to \mathbb R$ by ...
15
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4answers
746 views

How to prove $\int_0^{\infty}\frac{x^2+3x+3}{(x+1)^3} e^{-x}\sin x\, dx = \frac{1}{2}.$

I've just seen this integral crop up on another site and I can't see an obvious way to prove it. Any suggestions? $$\int_0^{\infty}\frac{x^2+3x+3}{(x+1)^3} e^{-x}\sin x\, dx = \frac{1}{2}.$$
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2answers
42 views

Simple Integration Question - Integral of the product of a function and its derivative.

Why does $\int y'(x) y(x) dx = y(x)^2/2 + C$? This seems to be true, at least according to Wolfram, but I do not understand how it is derived.
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4answers
136 views

Evaluate $\int_{-\infty}^\infty \frac{1}{(x^2+1)^3} dx$

Evaluate $\int_{-\infty}^\infty \frac{1}{(x^2+1)^3} dx$ I wasnt exactly sure how to approach this. I saw some similar examples that used Cauchy's theorem.
5
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1answer
109 views

Estimating $\int_0^x f(x-t)f'(t)dt$

I'm attempting to estimate $\int_0^x f(x-t)f'(t)dt$ in terms of a simple asymptotic expression with an error term for some 'well-behaved' functions, namely $f = O(x)$, of class $C^1$ or higher, with ...
0
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1answer
162 views

Evaluating $ \int \frac{1}{\sin x} dx $

Verify the identity $$\sin x = \frac {2 \tan\frac{x}{2}}{1 + \tan^2\frac{x}{2}}$$ Use this identity and the substitution $t = \tan\frac{x}{2}$ to evaluate the integral of $$ \int \frac{1}{\sin x} dx ...
2
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2answers
105 views

evaluation of the integral of a certain logarithm

I come across the following integral in my work $$\int_a^\infty \log\left(\frac{x^2-1}{x^2+1}\right)\textrm{d}x,$$ with $a>1$. Does this integral converge ? what is its value depending on $a$ ?
6
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3answers
647 views

Difficult general integral definite 0 to 1

$$\int_{0}^{1} \log^2(x)\cdot x^{k+1} dx$$ I tried integration by parts but it leads to an extremely complicated computation, which didnt lead me anywhere. Then I tried differentiating the beta ...
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0answers
40 views

Question about compact sets in $\mathbb{R}$

Suppose I am given a function $f$ on $\mathbb{R}$ and I'm asked to show that $ \int_K f(x) \,dx < \infty$ for any compact set $K \subset \mathbb{R}$. Would it be enough to only consider the ...
0
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1answer
41 views

Finding arc length by approximating

Let's say that a have a smooth curve in 2D and I want to find it's lenght. I split the curve in sections like here: http://www.whitman.edu/mathematics/calculus_online/section10.03.html. The only ...
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4answers
102 views

Integrating $\int _0^1\sqrt{9\sin^2(x)+9\cos^2(x)+16}\ \mathrm dx$

What I'm trying to do is find the arc length of $r(x) = \langle 3\cos(x),4x,3\sin(x)\rangle$ when $0\le x\le 1$. After I derived the vector and found the magnitude of that derived vector, I came up ...
4
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0answers
156 views

Solving $\int\frac{x}{(x^2+x+1)^{\frac{1}{12}}}$

I'm trying to solve $$\int_0^1\frac{x}{(x^2+x+1)^{\frac{1}{12}}}\mathrm dx$$ To calculate it I first tried to calculate the primitive function. So let $$\int\frac{x}{(x^2+x+1)^{\frac{1}{12}}}\mathrm ...
1
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1answer
44 views

Line integral of second kind over a circle: $\int \frac{xdy - ydx}{x^2+y^2}$

I've just get stuck with some task of line integral: $$\int \frac{xdy - ydx}{x^2+y^2}\quad \text{ over} \ x^2+y^2=R^2$$ I understand that I need to use polar coordinates, and I have such thing: $$x ...
0
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2answers
65 views

Evaluate $\lim_{y\to\\+0} \int_{0}^{1} \exp\left(-\frac{\arctan x}{y}\right)\,\mathrm dx$

I am trying to evaluate the following $$\lim_{y\to\\+0} \int_{0}^{1} \exp\left(-\frac{\arctan x}{y}\right)\,\mathrm dx$$ Seems useful to bring limit under integral, but can't see the solution. ...
0
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2answers
153 views

Finding the volume bounded by surface $y^2=4ax$ and the planes $x+z=a$ and $z=0$

The problem is stated below: Let $V$ be volume bounded by surface $y^2=4ax$ and the planes $x+z=a$ and $z=0$. Express $V$ as a multiple integral, whose limits should be clearly stated. Hence ...
2
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1answer
190 views

Approximating an integral with another integral with finite limits

I came across the following integral in my work $$\int_{-\infty}^{\infty} \frac{\frac{1}{(1- \ \ 2 \pi j s \theta)^{m}}-1}{2\pi j s }\ e^{-2\pi j s\sigma^2}\ ds $$ Assuming $\theta,m,\sigma^2$ are ...
5
votes
2answers
94 views

Integrate a periodic absolute value function [duplicate]

\begin{equation} \int_{0}^t \left|\cos(t)\right|dt = \sin\left(t-\pi\left\lfloor{\frac{t}{\pi}+\frac{1}{2}}\right\rfloor\right)+2\left\lfloor{\frac{t}{\pi}+\frac{1}{2}}\right\rfloor \end{equation} I ...
1
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2answers
77 views

Calculate the mean of the normal distribution function $\frac1 {2\pi \sigma^2}exp[-\frac {(x-\mu)^2} {2\sigma^2}]$ by integration.

I know that it must be $\mu$ but I cannot get the answer. This is my attempt so far: Normal distribution function = $N(x)=\frac1 {2\pi \sigma^2}exp[-\frac {(x-\mu)^2} {2\sigma^2}]$ $$\langle ...
2
votes
3answers
123 views

Evaluate $\int_0^\pi \arctan(\cos x)\,\mathrm dx$

I need to Evaluate $$\int_0^\pi \arctan(\cos x)\,\mathrm dx$$ . I tried to make an exchage $t=\cos x$ and then take the integral by parts
3
votes
3answers
99 views

Help with $\int \frac{1}{(\sin x + \cos x)}$ [closed]

Kindly solve this question $$\int \frac{1}{(\sin x + \cos x)} dx$$ I reached up to $$\frac{(1+\tan^2x)}{1-\tan^2x + 2\tan x}$$
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1answer
65 views

Evaluate $ \int \frac{\tan(x)}{2+\sin(x)}dx $

How do you evalute this integral? $$ \int \frac{\tan(x)}{2+\sin(x)}dx $$
3
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3answers
89 views

Evaluate $\int_0^\infty e^{-x^2}\cos\frac{t}{x^2}\,\mathrm dx$

How to integrate $$\int_0^\infty e^{-x^2}\cos\frac{t}{x^2}\,\mathrm dx$$ where $t$ is a constant/parameter?
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2answers
49 views

Integral of $\dfrac{\cos(x)}{5+3\cos(x)}$

I was doing $$\int\!\mathrm{d}x \dfrac{\cos(x)}{5+3\cos(x)}$$ and using the substitution $\cos(\theta) = \dfrac{1-t^2}{1+t^2},\quad t = \tan\left(\dfrac{\theta}{2}\right)$ ...
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0answers
28 views

Multiple integral of iterated kernel

I am trying to implent Volterra equations using resolvent kernel.To do this, the iterative kernel $$K_i(x,y) = \int\limits_x^y K_1(y,t)K_{i-1}(t, x)dt. $$ should be calculated. However, it is not ...
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2answers
35 views

Are these the same? $\int_{-2}^2(2x^2-x)^4$ and $2\int_{0}^2(2x^2-x)^4$

$$\int_{-2}^2(2x^2-x)^4\,\mathrm dx\quad\text {and}\quad2\int_{0}^2(2x^2-x)^4\,\mathrm dx$$ I tried to solve this and got different answers, but the other problems that I did got the same answers. ...
4
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1answer
63 views

Doubt in solution for evaluating $\int_0^1\int_0^1\int_0^1(1+u^2+v^2+w^2)^{-2}du~dv~dw$.

I've a doubt in the answer for evaluating the following integral: $$\int_0^1\int_0^1\int_0^1(1+u^2+v^2+w^2)^{-2}du~dv~dw$$ solution: call this integral as $I$ .By symmetry we may compute it ...
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0answers
25 views

Approximate the inverse Laplace transform

I am struggling with an inverse Laplace transform for a long time! Assume we have a function $m(t)$ and its Laplace transform is denoted by $M(s)$. I have derived the expression of $M(s)$ by some ...
0
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0answers
32 views

Integral and limits

My lecturer made the following remark: $$\frac{1}{2\pi i}\int_{\sigma_{0}-i\infty}^{\sigma_{0}+i\infty} y^{s} \frac{ds}{s} \neq \lim_{T\rightarrow \infty} \frac{1}{2\pi ...
2
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1answer
28 views

Doubt on integrating $f$ on a given region .

I was asked to integrate the following function: $f(x,y,z)=1-z^2$ on $U$ where $U$ is a pyramid with the top vertex $(0,0,1)$ and base vertices :$(0,0,0),(1,0,0),(0,1,0), (1,1,0)$ the ...
3
votes
1answer
36 views

Is the following function Riemann integrable?

Consider $f:\mathbb{R^2}\to\mathbb{R}$ defined by: $$f(x,y):=\arctan\frac{1}{x-y}\quad\forall x\neq y$$ $$f(x,x):=0$$ Is the function Riemann integrable in the square $[0,1]\times[0,1]$? I just ...
1
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1answer
68 views

Find expected present value of a continuous payment stream

I have a question for the financial part of my course which I am struggling to answer as i am not sure my answer makes sense. Question: Time is counted from the present t = 0 in years. Suppose for ...
2
votes
1answer
74 views

use fundamental theorem of calculus to find a function $f(x)$ and a number $a$

I thought I understood the fundamental theorem of calculus but I'm confused on the following problem.. Use the Fundamental Theorem of Calculus to find a function $f(x)$ and a number $a$ so that ...
0
votes
1answer
42 views

Expectation of random varible with normal distribution composed with exponential [duplicate]

I am trying to find $\mathbb{E}(e^{-X})$ where $X$ is a random variable with a general normal distribution. I end up with $$(2\pi \sigma)^{-\frac{1}{2}} \int_{-\infty}^{\infty} ...
5
votes
1answer
188 views

Proof that $\int_{0}^{1}\frac{dx}{1+x^6}=\frac{\pi+\sqrt3\log(2+\sqrt3)}{6}$ without residues.

How do you prove that $$\int_{0}^{1}\dfrac{dx}{1+x^6}=\frac{\pi+\sqrt3\log(2+\sqrt3)}{6}$$ My steps: First sub $\displaystyle u=x^3, \sqrt[3]u=x, dx=\dfrac{u^{-2/3}}{3} ...
0
votes
1answer
44 views

Integration over a spherical surface

Suppose that we have let's say a function of "something" that is everywhere on the spherical surface zero except at one point it is finite. Why is the integral of such a function over the surface is ...