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
votes
0answers
22 views

Trapz vs gausslegendre - integration methods [closed]

I have a code that uses trapz. But i want to try gausslegendre, but its different method and its not easy to change. Any help ? ...
2
votes
2answers
44 views

bend measurement and calculating $\int_4^8 \sqrt{1+{\left(\frac{{x^2-4}}{4x}\right)^2}} $

How can i get the measure of this bend : $y=\left(\frac{x^2}{8}\right)-\ln(x)$ between $4\le x \le 8$. i solved that a bit according to the formula $\int_a^b \sqrt{1+{{f'}^2}} $:$$\int_4^8 ...
6
votes
1answer
113 views

Evaluating $\int\sqrt{\frac{1-x^2}{1+x^2}}\mathrm dx$

Evaluating $$\int\sqrt{\frac{1-x^2}{1+x^2}}\mathrm dx$$ I had read the similar problem, but it doesn't work.
0
votes
1answer
38 views

Complete the square doesn't work here.

In this substitution, I tried the method of complete the square, but at the end the answer is not correct. I'm totally sure that the answer is: ...
0
votes
0answers
16 views

Need Help With this Integral:

I am working on a probability problem and I have figured everything out except I am having trouble calculating this variance: $\int_{-\infty}^{\infty} (t-s)_+^d-(-s)_+^d ds$ and I was wondering if ...
1
vote
1answer
53 views

How is this trival?

I was reading an article today and on section 2 it is indicated that if we are given a Radon Measure $\mu$, and a real $p$ then fast convergence entails trivially almost sure convergence, where fast ...
0
votes
0answers
17 views

How to choose grid for a numerical integral of complex function?

I need to numerically integrate a complex function $f(x)$ on R, i.e. to approximate $\int_{-\infty}^\infty{f(\xi)d\xi}$. Performance is crucial as the integration is repeated a high number of times ...
0
votes
0answers
19 views

Does this theorem concerning upper and lower bnound of a monotone decreasing function have a formal name?

This is the theorem: Let $g$ be a monotone decreasing function and let $a,b \in \mathbb{N}$. Then the following holds true: $$\int_{a}^{b+1}g(x)dx \overset{(i)}{\leq} ...
10
votes
3answers
165 views

Calculating $\int_0^{\infty } \left(\text{Li}_2\left(-\frac{1}{x^2}\right)\right)^2 \, dx$

Do you see any fast way of calculating this one? $$\int_0^{\infty } \left(\text{Li}_2\left(-\frac{1}{x^2}\right)\right)^2 \, dx$$ Numerically, it's about $$\approx ...
6
votes
3answers
119 views

Closed-form of $\operatorname{Li}_2\left(1 \pm i\sqrt{3}\right)$

I've found the following identity while I was going through a quite difficult path. $$ \Re\operatorname{Li}_2\left(1 \pm i\sqrt{3}\right) = \frac{\pi^2}{24} -\frac{1}{2}\ln^2 2 - ...
0
votes
0answers
12 views

General case for differentiation under the integral sign

What is the most convenient way to decide if we can differentiate under the integral sign? If the integrant is a smooth function, could we do so?
14
votes
3answers
184 views

Prove that $\int_0^1 \frac{1}{1+\ln^2 x}\,dx = \int_1^\infty \frac{\sin(x-1)}{x}\,dx $

I've found the following identity. $$\int_0^1 \frac{1}{1+\ln^2 x}\,dx = \int_1^\infty \frac{\sin(x-1)}{x}\,dx $$ I could verify it by using CAS, and calculate the integrals in term of ...
3
votes
1answer
37 views

computing an integral (fraction of real powers)

I want compute the following integral depending on the real parameters $\alpha, \beta > 0$ and $C >0$ $$ \int_0^1 \frac{u^{2\beta}}{C+u^{2(\alpha+\beta)}} du$$ Thanks a lot for any clue !
1
vote
1answer
23 views

Find the values of the derivatives of the integral with a variable inside its limits.

$\require{cancel}$ Problem: I have the function $g: \mathbb{R} \to \mathbb{R}$ defined as $$ g(x)=\int^{(1+x^2)}_{-(1+x^2)} sin(t^3)\ dt,\ x \in \mathbb{R} $$ I would like to calculate values of ...
0
votes
4answers
38 views

why $\int\frac{dx}{2\sqrt x +2x} = \ln(1+\sqrt x)$+C [closed]

I don't know why $\int\frac{dx}{2\sqrt x +2x} = \ln(1+\sqrt x)$+C .The maple solve that like this : $$\int\frac{dx}{2\sqrt x +2x}:$$ $$=\int\frac{du}{u+1}$$ $$=\int\frac{du_1}{u_1}$$ $$=ln(u_1)$$ ...
-1
votes
0answers
24 views

How do we know which quantity to be integrated with respect to which? [closed]

Suppose I write z = x1y1 + x2y2 + ... + xnyn So, if x and y are not discrete values, but continuous functions, then would z be written as the integral of x.dy or would it be y.dx or something else ...
2
votes
1answer
48 views

Convergence of an integration $t=\int_{x_0}^{x_1}\sqrt{\frac{m}{2(E_0-V(y))}}dy$

When I am reading Brian Hall's "Quantum Theory for Mathematicians", I came across an integration (frequently appeared in physics textbooks) $$t=\int_{x_0}^{x_1}\sqrt{\frac{m}{2(E_0-V(y))}}dy.$$ The ...
4
votes
4answers
57 views

Integration using trig substitution or substitution

I was trying to review calculus integration techniques before my differential equations class. I came across $\int \frac{1}{\sqrt{1-2x^2}}\,\mathrm{d}x$. I can't exactly figure out a good way to solve ...
2
votes
5answers
105 views

definite integral of $x^2e^{-x^2}$

I am trying to calculate the integral of this form: $\int_{-\infty}^{+\infty}e^{-x^2}\cdot x^2dx$ I am stuck. I know the result, but I'd like to know the solution step-by-step, because, as some ...
3
votes
1answer
35 views

Surface of $(x^2 + y^2 + z^2)^2 = a^2 * (x^2 - y^2)$ using surface integrals

I have to find the surface of $$(x^2 + y^2 + z^2)^2 = a^2(x^2 - y^2)$$ using a surface integral and really have no idea what to do... I would really appreciate it if you could give me an idea.
-3
votes
0answers
72 views

$\int \left(\cos\left(e^{\sin(x)}\right)\right) \text{d}x$ [closed]

I have to find this integral but I have no idea if there is any solution to it: $$\int \left(\cos\left(e^{\sin(x)}\right)\right) \text{d}x$$
1
vote
1answer
62 views

How was this differentiated?

How red-circled function with 1/D is equal to green-circled? Note: D is equal to dy/dx. Update: Complete pic
0
votes
1answer
35 views

Line integral and checking its path independence in three dimensions

I have a following exercise, falling under the topic of line integrals. Calculate the integral: $$I=\int\limits_{\gamma} \sin(yz)\,dx+xz\cos(yz)\,dy+xy\cos(yz)\,dz$$ Where: ...
3
votes
4answers
99 views

Why $\int_{0}^{\pi/2}\tan(x/2) dx= \ln 2$ [on hold]

I don't know why $$\int_0^{\pi/2}\tan\frac{x}2\ dx= \ln 2.$$ How can i solve this to get that answer?
3
votes
2answers
140 views

Proving that $\int_0^{\pi/2} (\sin (\tan (x))+\cot (x) \cos (\tan (x))-\cot (x))\cot (x) \, dx=\frac{\pi(e-2)}{2e}$

I think one of the ways of doing it is by the use of the differentiation with parameter. Do you see an easy way of calculating it by real methods? $$\int_0^{\pi/2} (\sin (\tan (x))+\cot (x) \cos ...
0
votes
0answers
23 views

How to prove $\int_{\Omega} \sum_{i=1}^{N} f_i(x)dx$ is equivilant with $\sum_{i=1}^{N} \int_{\Omega} f_i(x)u_i(x)dx$

I have a 2D image in $\Omega$ space. Assume that the space can be separated into $N$ sub-regions $\Omega_i$ such that $\Omega_i \cap\Omega_j=\emptyset$; $\Omega_i \cup \Omega_j=\Omega, \forall ...
-1
votes
0answers
15 views

Why it is the same when a curve is rotated 4 right angles or 2 right angles about the y-axis? [closed]

i realise when the question ask for volume of revolution form when rotated 2 right angles or 4 right angles about the y-axis, both yield the same result. however, this is not so when rotated about ...
0
votes
0answers
23 views

Question on integration (volume) [closed]

When a curve is rotated through 4 right angle about the x-axis, it is taken to be rotating through $2\pi$, which makes sense as $2\pi$ is $360$ degrees. But when it is rotated through y-axis, its is ...
1
vote
1answer
58 views

Integration of a analytic function

here is the problem I currently try to solve: $$\int\limits_{-\infty}^{+\infty}\left((1+ixa^2)^{-\frac{n_1}{2}}\cdot(1+ixb^2)^{-\frac{n_2}{2}}\right)e^{icx} \mathrm{d}x $$ with $a,b,c\geq0$ (real ...
0
votes
2answers
43 views

Evaluate $\int _{|z|=2}ze^{\frac{3}{z}} dz$

How to integrate this $\int _{|z|=2}ze^{\frac{3}{z}} dz$ I tried to do this by substituting $z=2e^{i\phi};0\leq \phi\leq 2\pi$ But cannot proceed further.Any help
4
votes
3answers
575 views

How can I integrate this expression? [closed]

$ \displaystyle\int \sqrt{\dfrac{x^3-3}{x^{11}}}\,dx$ It seems like substitution could not do it. Is there another way?
12
votes
6answers
896 views

Where did I go wrong in my evaluation of the integral of cosine squared?

$$\int{\cos^2(x)}dx$$ Where did I go wrong in my evaluation of this integral? $$=x\cos^2x - \int-2x\sin(x)\cos(x)\,dx$$ $$=x\cos^2x + \int x\sin(2x)\,dx$$ $$=x\cos^2x + \left(\frac {-x\cos(2x)}2 ...
5
votes
3answers
151 views

Integrating and indefinite integral any possible way

How do I integrate the following: $$\large \displaystyle\int_0^{\infty}\frac{x^4e^x}{(e^x-1)^2} \, dx$$ I have tried everything from integrating by parts to simply expanding the denominator, but it ...
1
vote
1answer
21 views

Average integral for continuous functions with compact support

Let $f$ be a continuous function with compact support in $\mathbb{R}^n$. Show that \begin{equation} \lim_{r\to 0} \frac{1}{|B_r(x)|} \int_{B_r(x)} f(y)\,dy = f(x), \end{equation} where $B_r(x)$ is the ...
4
votes
0answers
68 views

On finding an explicit form of a particular recurrence relation

Let $f$ be integrable over the interval $[0, 1]$, and $$I_n = \int_0^{1} x^n f(x) \, \mathrm{d}x.$$ Suppose $f(x) = f(1-x)$; we can then show that $$I_n = \sum_{k=0}^{n} \binom{n}{k} (-1)^k \, ...
3
votes
3answers
107 views

I don't know why $\int_{-3}^{-2} \frac{dx}{x} = \ln(\frac{2}{3})$

I don't know why $$\int_{-3}^{-2} \frac{dx} x = \ln \frac{2}{3}.$$ How can i solve this to get that answer?
1
vote
1answer
29 views

Vector valued integral in spherical coordinates

Whenever I have been presented with integrals as (*), I have always used some sort of symmetry to get around actually calculating the integral. Now it just struck me that I have no idea how to ...
0
votes
2answers
89 views

$\int f(x)h(x)d x = \int g(x)h(x)dx$ for any bounded integrable function $h$ if and only if $f(x)=g(x)$

How to show that $\int f(x)h(x)d x = \int g(x)h(x)dx$ for any bounded function $h$ if and only if $f(x)=g(x)$. The "if" part is clear. How to show the only if part?
-1
votes
0answers
38 views

Integrate with respect to $x$ (vertical strips) to calculate the volume generated when the region is rotated about the vertical axis $x = −2$. [closed]

the equations that are used are $y=x$, $y=x^3$, $y=8$, $x=9$. i have done the other question (Integrate with respect to $y$ (horizontal strips) to calculate the area of the region) and found it to be ...
1
vote
1answer
40 views

Difficulties of convergence of the partial sums of the Fourier inversion formula

Define the Fourier inversion formula over $\mathbb{R}^n$ by $$ f(x)=\int_{\mathbb{R}^n}e^{2\pi ix\cdot\xi}\hat{f}d\xi $$ where $\hat{f}$ is the Fourier transform over $\mathbb{R}^n$ $$ ...
0
votes
3answers
57 views

Why is this integral “inconsistent”?

I am currently working on the following integral: $$\int_{-\infty}^{\infty} xe^{-|x+1|} dx$$ The method I use for it is Integration by Parts. When I calculate the integral $\int_{-\infty}^{\infty} ...
-2
votes
4answers
51 views

How to integrate $\frac{\sec x}{ \tan^2 x}$ [closed]

How to integrate $\frac{\sec x}{ \tan^2 x}$ , already tried dividing it first but am not getting it
0
votes
0answers
33 views

Fourier series for a Sinusoid in a conventional way?

So my TA in class introduced this amazing way of finding fourier series coefficients for a sin wave, by writing $ sin( \omega t ) = (e^{i\omega t}-e^{-i\omega t}) / 2i $ ----(1) Hence getting the ...
0
votes
1answer
67 views

Any suggestions on how could I solve this integral involving a square-root of a polynomial in the denominator?

This is not a homework problem, so there is no guarantee that this integral is solvable analytically. $$ \int_0^\infty \frac{x^2(1-x/2)}{\sqrt{x^2(1-x/2)^2+b}}dx\,. $$ It looks simple enough, but ...
0
votes
0answers
21 views

Integral equation solving methods

There is optimization problem, which is about unknown function $\varphi$ under integral sign: $\iint\limits_{[a, b] \times [c, d]}^{} K(x, y) \varphi (x, y) dxdy \to \max$ where is $\varphi (x, y) \in ...
2
votes
2answers
50 views

Find the Integrating Factor

Show that the differential equation $(1-2x^2y^2-4xy^3)dx + (2-2x^3y-4x^2y^2)dy=0$ is not exact, but admits integrating factor $\mu=\mu(xy)$. Find $\mu$ and solve the equation. With the method I ...
1
vote
3answers
21 views

Transformation to polar coordinates in an integral

Suppose that the domain of integration for a double integral is: $\{(x,y), - \infty < x \le a, -\infty < y \le a \}$. If I want to do a change of variable (to polar coordinates), how do I ...
4
votes
2answers
90 views

Double integral with a product of dilog $\int _0^1\int _0^1\text{Li}_2(x y) \text{Li}_2((1-y) x)\ dx \ dy$

One of the integrals I came across these days (during my studies) is $$\int _0^1\int _0^1\text{Li}_2(x y) \text{Li}_2((1-y) x) \ dx \ dy$$ that can be turned into a series, or can be approached by ...
2
votes
2answers
29 views

Let $f:\Omega\to(0,+\infty)$ and $\ln(x)$ be $\mu$-integrable

Show that $\displaystyle \lim\limits_{p\to 0^+} ||f||_p = \exp(\int\ln(f)\,d\mu)$. In case it comes to be helpful. So far I've shown that $\displaystyle\lim\limits_{p\to ...
2
votes
1answer
63 views

Integral Evaluation.

How can we justify the fact that some integrals can't be evaluated? It's like we can't sum up a function within two bounds or we are unable to find the area under the curve of a function. How's that ...