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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0
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2answers
117 views

Integrate along the vertical strip

I want to show that some integration with vertical line is bounded. function $f(\mu)$ is given by $$ f(\mu)=A^{-\sqrt{\mu}} \frac{(B_1-\sqrt\mu)}{(B_2-\sqrt\mu)(B_3+\sqrt\mu)} $$ where $f$ is defined ...
0
votes
2answers
86 views

Integrating $\int \frac{\sin^{-1}(x)}{\sqrt{1+x}}dx$ by parts

I have a question that requires me to integrate the following by parts. I have done the question but apparently my answer does not match that of wolfram alpha's. $$\int ...
5
votes
3answers
135 views

for each $\epsilon >0$ there is a $\delta >0$ such that whenever $m(A)<\delta$, $\int_A f(x)dx <\epsilon$

This is an old preliminary exam problem: Show that, for every nonnegative Lebesgue integrable function $f:[0,1]\rightarrow \mathbb{R}$ and every $\epsilon>0$ there exists a $\delta>0$ such ...
5
votes
5answers
80 views

Properties of $L^2(-1,1)$ functions

I want to show that there is no function $v \in L^2(-1,1)$ with $\int_{-1}^{1} v(x)\phi(x) dx = 2\phi(0)$ for all $\phi \in C^\infty_0(-1, 1)$ ($\phi$ is $0$ everywhere but $[-1,1] $). I know about ...
1
vote
1answer
39 views

Evaluating the line integral

The question I'm working on states: Let $C$ be the curve in $\mathbb R$ consisting of line segments from $(4,1)$ to $(4,3)$ to $(1,3)$ to $(1,1)$. Let $F(x,y) = (x+y)i + (y-1)^3 * e^{\sin(y)}j$. ...
4
votes
1answer
113 views

How to prove $\int_{0}^{\infty}\frac{e^{-\left(\sqrt{x}-a/\sqrt{x}\right)^2}}{\sqrt{x}}dx=\sqrt{\pi},\,a>0$?

We know that $$\Gamma\left(\frac{1}{2}\right)=\int_{0}^{\infty}\frac{e^{-x}}{\sqrt{x}}dx=\sqrt{\pi} $$ but it seems that, for every $a>0 $ we have ...
2
votes
2answers
49 views

How to evaluate two integrals (double and triple)? One with a conic boundary and the other with a square root boundary.

Calculate the integral: $$\iiint_V z^2 dx dy dz$$ Where $V \subset R^3$, bounded by: $y = \sqrt z$ rotated around $Oz$, and the plane $z = h$ $(h>0)$. My try: Introduce the polar coordinates ...
1
vote
1answer
29 views

Support of convolution

Assume $u \in L^1(\mathbb{R}^n)$ and $\mathrm{ess\,supp}(u) \subset U,$ where $U$ is a bounded open set. Now we compute the convolution of $u$ with a function $\eta \in C(\mathbb{R}^n)$ with ...
1
vote
1answer
47 views

Difference of two definite integrals

$F:\mathbb{R}\to \mathbb{R}$ $$F\left(x\right)=\frac{\left(x^2+ax+5\right)}{\sqrt{x^2+1}}$$ Find $a$ so that $$\int _0^2\left(F\left(x\right)\right)dx-\int _{-2}^0\left(F\left(x\right)\right)dx=2$$ I ...
6
votes
1answer
123 views

How would I complete my proof that $\int_a^bf(g(x))\,dx = \int_{g(a)}^{g(b)}f(x)\frac{d}{dx}(g^{-1}(x))\,dx$?

Around two years ago during a second semester calculus class, my professor remarked that $\int\sin(x^2)\,dx$ could not be integrated. Being a bit defiant, I tried (in vain) to prove him wrong. While ...
-1
votes
3answers
87 views
0
votes
2answers
62 views

How to solve this type of integrals?

I have specific problem for solving this type of integral : $$\int \frac{x-y}{x^2-2y^2}\, dx $$ I tried applying partial fractions, making the denominator complete square but the sum keeps on ...
0
votes
2answers
28 views

Class of differentiable functions and Lipschitz continuity

I am reading lectures notes by Dr. Yuvi Nesterov's "Introductory Lectures on Convex Programming ". On page 25, Lemma 1.2.2, to prove $f''(x) \leq L$, (where $f(x) \in C_L^{2,1}(R^n)$, $L$ is Lipschitz ...
2
votes
3answers
58 views

The limit of the integral when the set is decreasing in probability to zero.

This is an exercise problem(#2 in section 3.2) from 'A course in probability theory'. If $E(\vert X \vert ) < \infty$ and $\lim_{n \to \infty} P(A_n) = 0,$ then $\lim_{n \to \infty} \int_{A_n} X \ ...
0
votes
1answer
43 views

Is the function $\ln (u(x))$ integrable when $u$ is bounded and positive?

Consider $\Omega$ an open bounded domain in $\mathbb{R}^n$ and $ u \in L^{\infty} (\Omega)$ a positive function. My question is : the well defined function $\ln (u(x))$ is integrable? Intuitively ...
1
vote
1answer
64 views

Computing a limes via integration

Currently, I am studying for an exam (topics: real analysis, integration etc.). I came across the following exercise: Let $\lambda>-1$ and let $(a_n)$ be the sequence defined by ...
0
votes
2answers
99 views

Is my answer correct for $\int \frac{\sin(x)}{\cos^3(x)} \;\mathrm{d}x$

I said $\frac{\tan^2(x)}{2} + c$ but my book says $\frac{1}{2\cos^2(x)} + c$
11
votes
7answers
918 views

Why does the “separation of variables” method for DEs work? [duplicate]

Heyho, I am using the separation-of-variables method for quite a while now, but what was always bothering me a bit, is why is it possible to do those operations. I'll give a concrete example (source ...
2
votes
1answer
84 views

Recursive Integral.

If $\int _o^1 e^x(x-1)^ndx=16-6e$ find the value of n (n is a positive integer $n\le5$). ATTEMPT: Let $I=\int _o^1 e^x(x-1)^ndx$ By using $\int _a^b f(x)dx=\int _a^b f(a+b-x)dx$ $I=\int _o^1 ...
3
votes
2answers
46 views

Limiting case of of integral.

Let $f(x)= \lim\limits_{n \to \infty}\dfrac{\cos x}{1+(\arctan x)^n} $ then, Evaluate $\int_0^\infty f(x)\,dx.$ ATTEMPT: Here $$I=\int _o^\infty \lim\limits_{n \to \infty} \frac{\cos x}{1 + ...
0
votes
1answer
16 views

Equivalent definitions of an orthonormal function

I want to prove that the following two definitions for an orthonormal function $\phi$, in terms of $kT$ time shifts, are equivalent. So let $T$ the symbol period and $k$ an integer. Definition 1 ...
0
votes
1answer
37 views

Integrating $\prod_{i=k+1}^{kN} \left(\int_{-\infty}^{\infty}dp_i\right)\times$ with conditions on $p_i$

I am trying to integrate this expression which came up in a derivation of the momentum distribution function for an ideal gas. $\Theta(x)$ is the Heaviside step function which is $1$ when $x$ is ...
16
votes
1answer
194 views

Is it possible to find indefinite integral of $\int \frac{1} {{\sin(x)+\sec^2(x)}}\mathrm{d}x$?

I am in standard $XI$ (i.e.11) and newbie in learning topic of integration. My friend asked me to find indefinite integral of the example shown below $$y=\int \frac{1} {{\sin(x)+\sec^2(x)}} \, ...
4
votes
4answers
244 views

Derivative of a definite integral issue

$g:\mathbb{(0,1]}\to \mathbb{R}$ We have the function $$g\left(x\right)=\int _x^1\left(\frac{\sin\left(t\right)}{t}dt\right)\:$$ Show that the function is strictly decreasing. So I thought that I'd ...
2
votes
0answers
51 views

What are the restrictions on using substitution in integration?

* One photo is equal 1000 words. * Integration done by substitution $u=\tan x$. Integration done by substitution $u=\tan {x\over 2}$. The source function is a continuous positive function ...
3
votes
2answers
137 views

Turning infinite sum into integral

Could someone explain how one can replace infinite sum with integral? Examples (or you may use your own, it doesn't matter as I want to understand principles): $$\frac{1}{n} \sum_{i=1}^{n} \sin ...
-6
votes
4answers
94 views

Easy integral with cube root [closed]

Can you help me? evaluate the following integral: $$\int_0^\frac {\pi} {2} \frac {\sqrt[3]{(sin^2x)}} {\sqrt[3]{(sin^2x)}+{\sqrt[3]{(cos^2x)}}}\,\mathrm{d}x$$ Give me show step by step solutions ...
0
votes
7answers
151 views

Evaluating the indefinite integral $\int \frac{2x+3} {{x^2-2x-3}}\mathrm{d}x$ [closed]

Evaluating an indefinite integral it is simple. Hello it is simple but I can not. Can you help me? evaluate the following integral: $$\int \frac{2x+3} {{x^2-2x-3}}\mathrm{d}x$$ Give me show step by ...
-2
votes
6answers
104 views

Evaluating an indefinite integral with a square root in the denominator

Hello it is simple but I can not evaluate the following integral: $$\int \frac{1} {\sqrt{x^2-2x-8}}\,\mathrm{d}x$$ Give me a clue or show step by step solutions please. Thank you very much.
1
vote
1answer
59 views

What does $d \| \vec{x} \|$ represent?

I am doing some review problems for my upcoming exam and something has come up that I don't understand. $$\int_{\gamma} f(x,y) \space d \|\vec{x} \|$$ $\gamma(t)=(\cos(t),\sin(t))$ ...
2
votes
0answers
86 views

Computation of integral on parametrized curve

if $U$ and $V$ are two open subset of $\mathbb{R}^{n}$, $\varphi:U\rightarrow V$ a $C^{1}$ diffeomorphism, then we have the change of variable formula for the Lebesgue integral: ...
0
votes
0answers
70 views

Continuous and Discrete summation

I have a continuous curve $f(x)$ say constant in interval $0$ to $P$ and area under the curve is unity. Alternately, $f(x) = 1/P$ for $ 0\le x \le P$. So I can calculate $G = \int_{0}^P ...
1
vote
0answers
68 views

Explanation of a Diagram in Wikipedia

In the following diagram (from Wikipedia), there are levels on $y$-axis by red lines. I didn't understand how the rectangles below these red line are drawn? Can one help me? I am thinking the area of ...
1
vote
2answers
62 views

Evaluating simple integrations according to Lebesgue

One difference between Riemann's and Lebesgue's approach of integrations is that Riemann partitions domain of integration, whereas Lebesgue partitions the range of the function. I will consider two ...
0
votes
1answer
44 views

What is the closed form of the given integral?

Does a closed form of integral for $$\int_a^b e^{-c \left( x+\frac{a}{x}\right)} dx$$ exist? How can I approximate such integral? Here, $a=0^+$ (greater than $0$) and c is a positive constant.
-1
votes
3answers
103 views

Integrate $\ln x \cos(\ln x) \,dx$ [closed]

$$\int \frac{\ln x\cdot \cos(\ln x)} {x}\,dx$$ How to calculate this integrate thank you very much
1
vote
2answers
64 views

integral question help me please [closed]

$$\int_0^1 \frac{f(x)} {f(x)+f(1-x)}dx$$ Thank you very much
2
votes
0answers
26 views

Definition of Integration of a Differential From in Lee's Introduction to Smooth Manifolds.

On pg. 402 of Lee's Introduction to Smooth Manifolds (Second Edition), the following is said to define the integral of a differential form on $\mathbf R^n$: Let $D$ be an open domain of integration ...
1
vote
1answer
176 views

a vector calculus problem when coping with problem 9 chapter 5 evans

Hi I was trying to understand the solution given by Ray Yang in the post question 9 - chap 5 evans PDE It gets to the sort of things I am quite bad at... When I get to the point ...
3
votes
2answers
99 views

Find an equivalent of this function,

a) $f$ continuous on $[0,1]$ such that $f(x)>0$. Find an equivalent of $$h(\epsilon) = \int_0^1 \frac {f(x)}{x^2 + \epsilon^2}dx$$ when $\epsilon$ goes to zero and when $\epsilon$ goes to ...
1
vote
2answers
122 views

Estimating the value of an improper integral numerically

My question is how can I estimate the value of an improper integral from $[0,\infty)$ if I only have a programming routine that gives me the function evaluated at 100 data points, or 100 values of ...
0
votes
2answers
54 views

Computing the centroid of a three-dimensional solid

I am trying to find the centroid of the solid bounded above by $x^2 +y^2 +z^2 = 12$ and below by $z = x^2 + y^2$. I am having some troubles evaluating one of the sub-integrals, specifically, the ...
1
vote
1answer
49 views

Surface area of the circle

I was told to calculate the surface area of the following circle by the integration method (monte carlo) $x^2 + y^2 = 1$ The area of this circle is determined by the following inequalities: $-1 ≤ x ...
-1
votes
3answers
79 views

How can i tackle this? $\int_2^3 {\rm}\frac{1}{x([ln(x))]^5}dx$

I admit to having challenges this calculus and integral how do i solve this? I have not tried anything because it is just difficult and i need to do more practice I will not give up. bare with me ...
1
vote
0answers
37 views

Integration of $\int_0^{\pi/2}{\frac{\sin^3x\,{\rm d}x}{\sin^3x+\cos^3x}}$ [duplicate]

I want to evaluate $$\int_0^{\pi/2}{\frac{\sin^3x\,{\rm d}x}{\sin^3x+\cos^3x}}$$ I think I have tried everything here: Substitution of $\sin x=t$ (and a couple more), using trigonometry identities ...
0
votes
1answer
51 views

Identities for integral expressions like $\frac{\int_0^\infty a(x)f(x)dx}{\int_0^\infty a(x)g(x)dx}$

I have some fairly complicated integrals to work with, and sometimes to divide by each other. $$\frac{\int_0^\infty a(x)f(x)dx}{\int_0^\infty a(x)g(x)dx}$$ Are there any identities that can be used ...
2
votes
2answers
56 views

Why Is My Distance Travelled Overshooting?

I have the following pascal code to determine position and velocity from acceleration: ...
1
vote
0answers
78 views

Calculation of double integral: $\iint_D e^\frac{x-y}{x+y}\,dx\,dy$

I'm trying to solve the following double integral: $$I=\iint_D e^\frac{x-y}{x+y}\,dx\,dy$$ where $$D=\{(x,y)\in\mathbb R^2: 0\le x\le 2,\,1-x\le y\le 2-x\}$$ Let $$\left\{ \begin{array}{c} u=x+y \\ ...
7
votes
3answers
322 views

Infinite Integral of Trigonometric Functions

I am interested in finding the Integral: $I = \int\limits_{0}^{\infty} \sin x \,dx$. Clearly going the conventional way $I = -\cos (\infty) + \cos(0)$ will not lead to a definite answer. However I ...
1
vote
2answers
82 views

Complex value for volume, using triple integrals

I'm trying to calculate the volume a hyperboloid, within $$z=0$$ and $$z+\frac 12 x-3=0.$$ The hyperboloid: $$x^2+\left(\frac y2\right)^2-z^2=5.$$ I calculated the projections on $xz$, $yz$, to use ...