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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10
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3answers
240 views

Definite integral - closed form: $\int_{0}^{\infty}\cos\left(x^{4} + 1 \over x^{2}\right)\,{\rm d}x$

I'm struggling with this definite integral: $$ \int_{0}^{\infty}\cos\left(x^{4} + 1 \over x^{2}\right)\,{\rm d}x. $$ Any help will be greatly appreciated.
1
vote
0answers
49 views

Measure theory integration question involving continuous function

Quick measure theory question. Given that $\Omega \subset \mathbb{R}^{n}$ and $f$ is continuous on $\Omega$. How would you show that if $$\int_{\Omega}f \, dx = 0$$ Then $f = 0$ everywhere? Thanks ...
2
votes
1answer
112 views

Exercise on Dominated convergence theorem

Consider the sequence $f_n=(-1)^n \frac{x}{\log(1+x)} \chi_{(0,1/n)}(x)$. Is it true that $$ \sum_n \int_X f_n d\mu= \int_X \sum_n f_n d\mu$$ with $ X=(0,1)$? I was thinking about using the ...
0
votes
0answers
27 views

How could I calculate displacement along a 2D polyline by integrating each dimension separately?

This question's field of application is GPS trajectory analysis, but I'll try to give it a more abstract mathematical treatment. Suppose the trajectory of an object in 2D plane is described by a ...
3
votes
2answers
202 views

Evaluation of another definite integral

I have a definite integral that I am trying to solve. Any hint or reference is urgently sought. , where $r$ is any positive integer while $\psi$ and $\nu$ are positive real numbers.
3
votes
2answers
45 views

Integration of a function + Mean value theorem.

I have a practice question that goes as follows: Let $$f: [0, 1] \rightarrow \mathbb{R} $$ be a continuous function . show that there exists c in [0,1] such that $$\int_0^1 f(x) dx = f(c)$$ I'm ...
3
votes
3answers
127 views

Does $\int _1^{\infty }\left(\sin \left(x^2\right)\right)dx$ converge or diverge? [duplicate]

I'm in need of some assistance regarding a question in my Calculus textboox: Find if the following converges or diverges without calculating the integral: $$\int _1^{\infty }\left(\sin ...
0
votes
1answer
23 views

Arc length $\gamma(t) = (30 \sin t, 30 \cos t, 50 \cos t)$

Calculate the arc length of the curve $$\gamma (t) = ( 30 \sin t, 30 \cos t, 50 \cos t)$$ My attempt: $L = \int | \gamma'(t) | dt = \int \sqrt{ 30^2 \cos^2 t + 30^2 \sin^2 t + 50^2 \sin^2 t} \;dt ...
0
votes
1answer
22 views

If a function $F$ is continuous and bounded in $[0,\infty)$, then $F$ is integrable?

I was wondering if it is true to say that: if a function is continuous and bounded in $[0,\infty)$, then $F$ is integrable?
1
vote
1answer
43 views

How prove this $\frac{1}{2\pi h}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{\frac{-i(p-p')x}{h}}x^n\varphi{(p')}dxdp'$

show that this integral: $$\dfrac{1}{2\pi h}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{\dfrac{-i(p-p')x}{h}}x^n\varphi{(p')}dxdp'=\left(ih\dfrac{\partial }{\partial ...
0
votes
1answer
32 views

Solution for divergent integral

The integral $I = \int\limits_0^\infty \mathrm{d}x \, x \sin(x)$ does not converge. In physics we often use the principle of a convergence generating factor, in this example $I = ...
4
votes
1answer
60 views

Trigonometric substitution for integral question.

I'm reviewing my quizzes to study for midterm tomorrow, and I came across a problem where I'm supposed to integrate: $$\int\frac{1}{x^2\sqrt{4-x^2}}dx$$ I used Mathematica to solve the problem and ...
1
vote
1answer
73 views

Line integral + Work

$F=(z-y)i+(x-z)j+(2y-x)k$ Let $C$ be a curve formed by an intersection of the plane $2x-z=0$ with the cylinder of elliptical cross section $x^2+(y^2)/9=1$, assuming $y$ is parametrized along $C$ via ...
10
votes
1answer
323 views

Integral$\int_1^\infty \log \log \left(x\right)\frac{dx}{1-x+x^2}=\frac{2\pi}{\sqrt 3}\left(\frac{5}{6}\log (2\pi)-\log \Gamma \frac{1}{6}\right)$

UPDATED Hi I am trying to prove the following$$ I:=\int_1^\infty \log \log \left(x\right)\frac{dx}{1-x+x^2}=\frac{2\pi}{\sqrt 3}\left(\frac{5}{6}\log (2\pi)-\log \Gamma \big(\frac{1}{6}\big)\right). ...
4
votes
1answer
110 views

Integrating $ \int_0^1 \frac{e^{ix}}{x^6+1}dx $

Integrating $$ \int_0^1 \frac{e^{ix}}{x^6+1}dx $$ but having trouble. I factored $x^6+1$ but does not work for the problem. I used identity $e^{ix}=\cos x +i\sin x$, but got nowhere. I can say ...
3
votes
2answers
101 views

Integral of exponential using error function

I'm trying to solve some integrals below $$\int_{-\infty}^{\infty} {x^n e^\frac{-(x - \mu)^2}{\sigma^2}}dx$$ I am interested in the solutions where n = 0, 1, 2, 3, 4. I have learned that ...
0
votes
3answers
58 views

How to evaluate the integral of $(2x+e^{-x})^2$ from $x=0$ to $x=2$ [closed]

I have tried for some time to evaluate $$\int_0^2 (2x+e^{-x})^2 dx $$ but I have unfortunately beeen unsuccessful. Could somebody be of help?
1
vote
1answer
30 views

Conversion to polar coordinates of integral

Conversion to polar coordinates of this integral: $$\int_0^\sqrt{2} \int_y^\sqrt{4-y^2}\frac{1}{\sqrt{1+x^2+y^2}}\ dxdy $$
4
votes
3answers
121 views

Evaluating the integral $\int_{-1}^{1} \frac{\sin{x}}{1+x^2}dx$

I was asked to evaluate the integral $$\int_{-1}^{1} \frac{\sin{x}}{1+x^2}dx$$ if it exists. This is a problem from Calculus and the student has been taught how to use trigonometric substitution. ...
1
vote
1answer
28 views

Integrability and exponential integrability

I'm working on a paper, and I don't know if there is some kind of typo or if I just don't get what seems obvious to the author. Note : I'll be working with probabilities, but I guess this would be ...
1
vote
3answers
50 views

Let $(f_n)_{n\in\mathbb{N}} \rightarrow f$ on $[0,\infty)$. True or false: $\lim_{n\to\infty}\int_0^{\infty}f_n(x) \ dx = \int_0^{\infty}f(x) \ dx.$

The Assignment: Let $(f_n)_{n\in\mathbb{N}}$ converge uniformly to $f$ on $[0,\infty)$ and let the improper integrals of $f_n$ and $f$ exist on $[0,\infty)$. True or false: ...
1
vote
1answer
26 views

Indefinite integral convergence

How can I prove that this integral converges? $$ \int_0^1 \sqrt{\frac{1-kx^2}{1-x^2}} dx\quad\quad 0\leq k<1 $$ Edit: fixed typo dt -> dx
11
votes
4answers
423 views

Convergence $I=\int_0^\infty \frac{\sin x}{x^s}dx$

Hi I am trying to find out for what values of the real parameter does the integral $$ I=\int_0^\infty \frac{\sin x}{x^s}dx $$ (a) convergent and (b) absolutely convergent. I know that the integral ...
1
vote
2answers
76 views

How to integrate this fourier transform?

I want to integrate $$\int_{-\infty}^{\infty} \frac{e^{itx}}{{1+x^2}} dx.$$ I don't see how substitution or integration by parts could help here. Does anybody know how to do this?
1
vote
1answer
55 views

Indefinite integrals and changing variables

Why is it legal to change the variable of an indefinite integral? Consider $$\int \dfrac{dx}{\cos x}$$ If one were to say, $\text{Let } u=\cos x$, do we not now technically have ...
0
votes
0answers
48 views

First variation of convolution of two nonlinear functions, how to reexpress $\left[x \delta x * x^2 \right]$?

A new variational principle is presented in this paper: Mixed Convolved Action When trying to derive something like the equation of motion of a Duffing oscillator, I take the following approach: Set ...
2
votes
1answer
97 views

Integration in PID controller

I am trying to understand how come there is a phase difference is from the error signal and the output of my PID controller which consisting of I = 1. As far i've understood should the integration ...
3
votes
2answers
122 views

Evaluate the definite integral $ \int_{-\infty}^{\infty} \frac{\cos(x)}{x^4 +1} \ \ dx $

I am having more trouble with this problem then I feel like I should be. I set $ \ \cos(x) = e^{ix} \ $ and $ \ x^4 +1 = e^{\pi i /4} \ $ or $ \sqrt{i} \ $. I think I am suppose to do a residue to ...
2
votes
0answers
33 views

Is little-o preserved under integration and derivation of another variable?

Given an integrable function $g:\mathbb{R}\longrightarrow\mathbb{R}$, and a function $f:\mathbb{R}^2\longrightarrow\mathbb{R}$ such that $f(x,y)=o(x^{-1})$ when $x\rightarrow\infty$, i.e. ...
-1
votes
1answer
49 views

Strange Integral Notation? [duplicate]

When reading about an certain algorithm (about parameter estimation for Kalman Filtering page 7 eq 57) I found this notation: $\int dx f(x)$ which is normally written as $\int f(x) dx$. I spent a ...
1
vote
4answers
62 views

How to calculate this limit involving an integral

I missed a couple of classes so I'm having trouble doing this (and other similar) excercises of homework: $\lim\limits_{x\to0^+}\frac{\displaystyle\sqrt{x}-\displaystyle\int_0^\sqrt{x} ...
3
votes
2answers
76 views

Trigonometric integral with the function- sin

I need some help with this integral please: $\displaystyle\int x\sin\frac{1}{x}dx$
1
vote
2answers
63 views

Area calculation of ellipse $x^2/2+y^2=1$

Calculate the area of the ellipse that you get when you rotate the ellipse $$\frac{x^2}{2}+y^2= 1$$ around the x-axis. My approach has been to use the formula for rotation area from $-2$ to $2$. But ...
3
votes
1answer
67 views

$\int_0^1 e^{x^2}(2x-a)dx=0$

$\int_0^1 e^{x^2}(2x-a)dx=0$ where $a$ is any real number then predict the range of $a$ or find its value. My approach: $\int_0^1 e^{x^2}2xdx=(e-1)$ Can't integrate $a\int_0^1 e^{x^2}dx$.
8
votes
6answers
318 views

Calculus Question: $\int_0^{\frac{\pi}{2}}\tan (x)\log(\sin x)dx$

Can anyone help me to find $\int_0^{\frac{\pi}{2}}\tan (x)\log(\sin x)dx$? Any help would be appreciated. Thanks in advance.
6
votes
3answers
65 views

Evaluating integral with $\sqrt[3] x$ and $\ln x$

I found this example amongst set of examples to get ready on exam $$ \int\frac{\ln x}{\large\sqrt[3]{x}}\ dx $$ I am able to see the basic substitution, but I don't know how to count if further… ...
4
votes
2answers
48 views

Evaluating $\int \sqrt{x}\ln(2x)\:\mathrm{d}x$

I am learning to an exam and stumbled upon this integral. $$\int \sqrt{x}\ln(2x)\:\mathrm{d}x$$ the result should be $\frac{2}{3}\sqrt{x^{3}}\left(\ln(2x)-\frac{2}{3}\right)+C$, but I am still ...
0
votes
1answer
50 views

I stuck with this integral (lnx)

Good morning, I would like to ask you guys with helping me out with this integral, still can't get to the proper result...
3
votes
4answers
46 views

Per partes integration

Can I asked you guys for help with solving of this integral? Thank you. It could go well with the per-parted method, but I am not able to finish it. $$\int x^2 \ln x dx$$
1
vote
0answers
58 views

[Measure theory]. Proof of inequality of integrals of simple function

I have a question regarding a proof in my textbook. The theorem is as follows : if {$f_n$} is an increasing sequence of non-negative simple functions and $lim_{n \rightarrow \infty}f_n(x) \geq g(x), ...
8
votes
2answers
116 views

Evaluate $\displaystyle\int_0^\pi \frac{x}{1+\sin^2x} \ dx$

How can one evaluate $$\int_0^\pi \frac{x}{1+\sin^2x} \ dx\ ?$$
1
vote
1answer
50 views

Complex integration where the limits are complex numbers

I've been reading about integration in $\Bbb C$, and things look pretty similar to multivariable integration, however I found a series of excersices that baffled me a little, I don't know how to solve ...
0
votes
1answer
32 views

Find the volume of a cone whose length of its side is $R$

How can i compute the volume of a cone whose length of its side is $R$ and the vertex of the cone forms an angle $2θ$ . The top cone is a cap of a sphere of radius $R$. I tried to solve first in 2 ...
1
vote
0answers
34 views

Evaluating Surface Integrals

I've been told that surface integrals can be evaluated through projection by $$\iint_S f(x,y) \, dS = \iint_R \frac{1}{\| \frac{\mathbf{n}}{\|\mathbf{n}\|} \cdot \mathbf k \| }f(x,y) \,dx\,dy$$ ...
9
votes
4answers
201 views

Evaluate $\int_0^\infty \frac{(\ln x)^2}{x^2+4} \ dx$ using complex analysis.

Evaluate $\int_0^\infty \frac{(\ln x)^2}{x^2+4} \ dx$. This is the last question in our review for complex analysis. Hints were available upon request, but being the student I am, I waited until the ...
0
votes
2answers
100 views

Trouble with Indefinite Integral - $\int \frac{2x^{3}+8x}{\sqrt{(x^{2}+4)}}dx$

$$\int \frac{2x^{3}+8x}{\sqrt{(x^{2}+4)}}dx$$ I'm having issues integrating this. I found a step through and I do not understand the method of substitution. Would anyone be able to give me a brief ...
1
vote
1answer
26 views

Substitution integral with sines and cosines

Could I ask you guys to help me out with this? $\displaystyle\int \frac{\sin x}{\sqrt{2+\cos x}}dx$ Thank you in advance
1
vote
0answers
49 views

Finding reasonable change of variables?

Let $R \subseteq R^{2}$ be the first quadrant region bounded by the curves $x^2 + y^2 =4, x^2 + y^2 =9, x^2 −y^2 =1$ and $x^2 − y^2 =4$. Use an appropriate change of variables to evaluate. ...
1
vote
2answers
60 views

$\int \text{sech}^4 x \, dx$

How we can solve this? $$ \int \text{sech}^4 x \, dx. $$ I know we can solve the simple case $$ \int \text{sech} \, dx=\int\frac{dx}{\cosh x}=\int\frac{dx\cosh x}{\cosh ^2x}=\int\frac{d(\sinh ...
2
votes
5answers
74 views

Integration with Euler number and sin2x

I found this example in a textbook: $$\int e^{\cos^2 x}\sin2x dx$$ There are also results, but I am not even close to that...