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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4answers
39 views

integrate 1/(x*(x^2-1)^(1/2))

$\int\frac{1}{x*{(x^2-1)}^{1/2}}dx=?$ Hi! I'm new to the website and I didn't learn math in English so I may make mistakes with terminology. I have given a math homework and it says the answer is ...
0
votes
2answers
44 views

use parseval's identity to evaluate the integral $ \int_{-\pi}^{\pi}\sin^4 xdx$

use parseval's identity to evaluate the integral \begin{equation} \int_{-\pi}^{\pi}(\sin x)^4dx\end{equation} I'm familiar with Parseval's identity which states that for each piecewise continuous ...
-2
votes
0answers
46 views

Can anyone integrate this?

Question is self explanatory. This is the one I couldn't do. Any ideas would be very helpful: Calculate $\displaystyle\iint_{D}(2x-1)e^{x^2+y}\,dx\,dy$ where $D = \{(x,y) \in \mathbb{R}^2 \ | \ x+y ...
4
votes
1answer
82 views

Feynman technique of integration for $\int^\infty_0 \exp\left(\frac{-x^2}{y^2}-y^2\right) dx$

I've been learning a technique that Feynman describes in some of his books to integrate. The source can be found here: ...
0
votes
1answer
70 views

How to solve this integral: $\int \frac{\sqrt{-x^2 - x + 2}}{x^2}dx$?

Question is self explanatory. I have an exam and our professor gave us questions. This is the one I couldn't do. Any ideas would be very helpful: $$\int \frac{\sqrt{-x^2 - x + 2}}{x^2}dx$$
0
votes
0answers
25 views

double integral to consecutive integrals

can someone explain to me how was this step done? I have little knowledge about this area of mathematics, as much as i can say the double integral of a function F over an area X is constructed by this ...
-4
votes
2answers
42 views

Calculate $\int_{0}^{1} \frac{x^3}{\sqrt{1+x^4}}dx$ [on hold]

Help with this integral, please! $$\int_{0}^{1} \frac{x^3}{\sqrt{1+x^4}}dx$$
5
votes
0answers
69 views

The bathroom integral!

A couple of days ago I came across the following integral: $$\int_{0.02}^{0.08} \frac{e^{\cos^2x}+\ln(1+x)}{10^{x^3}\arctan(\sqrt{x})}\,{\rm d}x$$ The funny thing is, I found this integral written in ...
0
votes
2answers
35 views

Integrating $\sin^3(x)/(2+\cos(x))$

I could use some help solving the following integral: $$\int \frac{\sin^3(x)}{2+\cos(x)} dx$$ So far I tried using the equality: $$\sin^3(x) = \frac{3}{4} \sin(x) - \frac{1}{4}\sin(3x)$$ which ...
1
vote
1answer
41 views

Evaluating $ \int_0^\theta \cosh(a\sin x) dx$

The integral below seems quite simple, but I couldn't find anywhere the result. $$ I = \int_0^\theta \cosh(a\sin x) dx$$ I tried to expand it into Taylor expansion series and successfully evaluate the ...
-4
votes
2answers
29 views

Need help with integral [on hold]

Can you please help me solve this integral: $$\int \frac {\text dx}{x^4+1}$$ Thank you.
0
votes
0answers
13 views

Different results on doing $\frac{\partial}{\partial y}\left(\int_r^y \frac{1}{\sqrt{y^2-s^2}} ds \right)$ in different ways

I have a confusion when trying to get the result of the expression below, $$ I = \frac{\partial}{\partial y}\left(\int_r^y \frac{1}{\sqrt{y^2-s^2}} ds \right). $$ All variables are real and $y>r$. ...
10
votes
1answer
104 views

Evaluate the double sum $\sum_{m=1}^{\infty}\sum_{n=1}^{m-1}\frac{ 1}{m n\left(m^2-n^2\right)^2}$

As a follow up of this nice question I am interested in $$ S_1=\sum_{m=1}^{\infty}\sum_{n=1}^{m-1}\frac{ 1}{m n\left(m^2-n^2\right)^2} $$ Furthermore, I would be also very grateful for a solution ...
4
votes
0answers
59 views

integrate $\int \frac{1}{e^{x}+e^{ax}+e^{a^{2}x}} \, dx$

I've been trying to integrate $$ \int \frac{1}{e^{x}+e^{\omega x}+e^{\omega^{2}x}} \, dx $$ where $\omega=e^{2i\pi/3}$ but to no avail. I've tried substituting in $u=e^{(1+\omega)x}$ but ended up ...
0
votes
2answers
35 views

Is the Fourier transform of a continuous and compactly supported function summable?

Let $\varphi$ defined on the real line be continuous and with compact support. What can we say about the summability of $\hat{\varphi}$? I've gone through some theorems such as Parseval's without ...
0
votes
1answer
21 views

Finding volume of a revolution

I want to find the volume of the revolution that occurs when the region bounded by $y = x^2$ and $y = 1$ is revolved around the line $y=2$. The problem is that it is not solid and I cannot understand ...
-1
votes
1answer
24 views

Find the centroid of the region under the graph of the function $ w(x) = 4.5 + a x^{3} $ between $ x = 0 $ and $ x = 5 $. [on hold]

I need to find the centroid to determine where the equivalent force is acting on the region under the graph of $ w $ between $ x = 0 $ and $ x = 5 $. The given information is $$ w(0) = 4.5 ~ ...
4
votes
2answers
84 views

Finding $\int\frac{\sqrt{1-t^2}}{1+t^2}dt$

I wanted to find $\int\frac{\sqrt{1-t^2}}{1+t^2}dt$, so I substituted $t=\sin\theta$ and got $\int\frac{\cos^2\theta}{1+\sin^2\theta}d\theta$; but I'm not sure what the best way to proceed from here ...
2
votes
0answers
23 views

double integration with the same variable

I have the integral that I want to resolve. To calculate the flux of the electric machine, I have the following formula: $v_s= R_s \cdot i_s + \frac{\Phi _s}{dt}$ where $v_s, i_s, \Phi _s$ are ...
0
votes
1answer
56 views

How to find the integral with $\sqrt [ 3 ]{ x } +\sqrt [ 4 ]{ x } $ in the denominator?

How to evaluate $$\int { \frac { 1 }{ \sqrt [ 3 ]{ x } +\sqrt [ 4 ]{ x } } } +\frac { \log { (1+\sqrt [ 6 ]{ x } ) } }{ \sqrt [ 3 ]{ x } +\sqrt { x } } dx$$ I'm not being able to make the right ...
0
votes
3answers
25 views

other form of uv notation of integration by parts

So integration by parts looks like this $\int u\, dv = uv - \int v\, du$ But I have often seen it like this: $\int uv\,dx = u \int v\, dx - \int (u'\int v dx )\, dx$ I cannot prove this. $ uv ...
4
votes
3answers
124 views

What do these symbol mean?

I always see these symbols and others like it when looking at really advanced maths. I have yet to learn anything about it. I was wondering if someone could explain briefly what they are used for. ...
4
votes
2answers
48 views

Integrate with $-d(x/y)$

Here's an integral which I encountered that uses some unfamiliar notation for me: $$\int-\frac{d(x/y)}{\sqrt{1+(x/y)^2}}$$ What does this mean? I don't have much of an idea. Edit: This problem is ...
1
vote
1answer
19 views

Finding the equation of a curve where the gradient is $ax + b$ at all points.

The gradient of a curve is $ax + b$ at all points, where $a$ and $b$ are constants. Find the equation of the curve given that it passes through the points $(0,4)$ and $(1,3)$ and that the tangent at ...
5
votes
4answers
103 views

Finding $\int\frac{1}{x^{11}+4x^6}dx$

I wanted to find out if there is an easy way to evaluate $\displaystyle\int\frac{1}{x^{11}+4x^6}dx$. I substituted $u=x^5$ and then used partial fractions, but maybe there is a simpler way to find ...
-1
votes
0answers
14 views

What would be line integral along path number (iv) [on hold]

In the above image what should be the line integration along path iv. Thanks.
2
votes
0answers
36 views

Closed form of an infinite series of integrals $\int_{0}^{\eta} \cos nt \cos t \sqrt{\cos^2 t - \cos^2 \eta}$

Let $$ I(n,\eta) = \int_{0}^{\eta} \cos nt \, \cos t \, \sqrt{\cos^2 t - \cos^2 \eta}\; dt $$ where it is known that $0 < \eta \leq \frac \pi 2$. Is it possible to evaluate $S$, the infinite ...
-2
votes
0answers
18 views

Surface Integral of cone [on hold]

How would i calculate the surface integral of this the radius of the cone being 28.25 Thank you
13
votes
3answers
155 views

Integral involving Clausen function ${\large\int}_0^{2\pi}\operatorname{Cl}_2(x)^2\,x^p\,dx$

Consider the Clausen function $\operatorname{Cl}_2(x)$ that can be defined for $0<x<2\pi$ in several equivalent ways: ...
1
vote
1answer
41 views

Changing argument into complex in the integral of Bessel multiplied by cosine

I got a problem solving the equation below: $$ \int_0^a J_0\left(b\sqrt{a^2-x^2}\right)\cosh(cx) dx$$ where $J_0$ is the zeroth order of Bessel function of the first kind. I found the integral ...
2
votes
5answers
357 views

Creative way to find this area

Let's say We have a circle with center at $(0,0)$ with radius $r$ and we have the line $y=a$ where $0 \leq a \leq r$. the question is what is the area that between the circle and the line $y=a$(the ...
0
votes
0answers
47 views

Double Integration Working Help

Help I dont know how to approach this question, I have the answer but dont know how to write a detailed working process of obtaining it. It is supposed to find the surface area of a cone that is $z = ...
1
vote
1answer
46 views

Computing $\int_{\partial S} \frac{1}{1+z^n} dz$

Let $S=\{re^{it} : 0<r<R, 0< \varphi < 2\pi/n\}$ for some $R>1$ and $n\geq 2$. How can we compute $$\int_{\partial S} \frac{1}{1+z^n} dz?$$ I can't compute it directly, so I assume I ...
1
vote
1answer
48 views

Struggling to prove inequality

I've been given to following inequality to prove: (The hint given was not to evaluate the integral) \begin{equation*} \frac{1}{4} \leq \int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{sin(x)}{x}dx\leq ...
2
votes
2answers
63 views

what will be the value of this integral

$$ \large{ \int^{\Large{\frac{\pi}{2}}}_{0} \left[ e^{\ln\left(\cos x \cdot \frac{d(\cos x)}{dx}\right)} \right]dx}$$ We know that $\large{a^{log_a(c)} = c}$. But in this question, the expression in ...
1
vote
1answer
38 views

If $\Omega\subseteq\mathbb{R}^n$ is bounded, then $\int_\Omega|x-y|^{1-n}\,d\lambda < \infty$

Let $\Omega\subseteq\mathbb{R}^n$ be bounded with $n\ge 2$ $\left|\;\cdot\;\right|$ be the euclidean norm $\lambda$ be the Lebesgue measure on the Borelian $\sigma$-algebra of $\mathbb{R}^n$ I ...
3
votes
2answers
87 views

How to evaluate $\int_0^1 \ln(\frac{1+x}{1-x}) \frac{dx}{x} = \frac{\pi^2}{4}$?

Can anyone suggest the method of computing $\int_0^1 \ln(\frac{1+x}{1-x}) \frac{dx}{x} = \frac{\pi^2}{4}$ ? My trial is following first set $t =\frac{1-x}{1+x}$ which gives $x=\frac{1-t}{t+1}$ ...
2
votes
4answers
62 views

finding $\int {(2x + 5)^2}$

After slowly getting the hang of differentiation I have moved onto integration and I can't seem to understand this one. I know the answer is $$\frac{4x^3}{3} + 10x + 25x + C$$ I understand that ...
1
vote
3answers
48 views

Change the order of expectation

Sorry this might be a silly question, but I'm kind of confused and really want to make sure I'm correct. Let $v_1,v_2,\dots,v_n$ be $n$ i.i.d. random variables with the same range of ...
1
vote
0answers
43 views

Integral tending to an integral for $\pi$

I am examining: $$\int_0^1 (1-ax)^{1/2} dx$$ If we differentiate: $$\dfrac{d}{dx} \left[\dfrac{-2(1-ax)^{3/2}}{3a}\right]$$ we get to the function in the integral. The idea now is consider various ...
1
vote
0answers
10 views

Quadruple integral of the solution to a new type of fractional differential equation

Let $\text{D}$ denote the differential operator, and $\text{D}^n$ the $n$th application of $\text{D}$ (i.e. the $n$th derivative) for any positive integer $n$. Note that $\text{D}^0 = ...
-1
votes
3answers
46 views

find the derivative of the integral

Prove that the following integral $F(x)$ is differentiable for every $x \in \mathbb{R}$ and calculate its derivative. $$F(x) = \int\limits_0^1 e^{|x-y|} \mathrm{d}y$$ I don't know how to get rid of ...
9
votes
0answers
86 views

The Laplace transform of $\frac{\ln(1+at)}{1+t}$

By expressing the square of the exponential integral as a double integral and then making a change of variables, one can show $$ \int_{0}^{\infty} e^{-2zt} \ \frac{\ln(1+2t)}{1+t} \, dt = \frac{e^{2z} ...
-7
votes
1answer
45 views

Solve the integral $\int _0^1\:\frac{\sqrt[3]{x}+1}{1+x}dx$ [on hold]

Solve the integral $$\int _0^1\:\frac{\sqrt[3]{x}+1}{1+x}dx$$
1
vote
1answer
30 views

Stochastic Integral basics

As far as I understand, the stochastic integral is defined so that we can make sense of something like this: \begin{equation*} X_t = x_0 + \int_0^t g(s) ds + \int_0^t f(s) dW(s) \end{equation*} ...
1
vote
0answers
26 views

Integral Over the N-Sphere in the framework of chains:

Integration over manifolds is commonly defined with object called chains. What about if I want to integrate the exterior derivative of a $k-form$ over the n-sphere and use Stokes theorem: ...
-2
votes
2answers
70 views

Unable to understand integration formula $\int 2x dx=x^2+2$ [on hold]

$$\int 2x dx=x^2+2$$ How? In all books this step is done directly. I am a beginner in integration. Please explain with all the necessary steps.
0
votes
0answers
28 views

Functional Analysis, a question that needs clarification.

Find the norm of the linear operator $A:C[-1,1]\to L^p[-1,1]; p\geq1$ that is defined as: $$A(x(t))=\int_{-1}^{1}{{x(s)\over (t-s)^{1 \over 3}}}ds$$ Can someone provide an answer with a little more ...
1
vote
1answer
18 views

integrability of discontinuous functions

The FTOC states that if $f$ is continuous on $[a,b]$ then it is integrable. If $f$ is not defined at certain points of $[a,b]$ we can often give meaning to an improper integral. But under what ...
0
votes
0answers
27 views

Cauchy's Integral Formula Question- Calculating an integral with z^4 + 16 on the denominator

I think the first part of this question is okay. For the second part, I have found the roots and then calculated the absolute difference between these roots and i and, as they are all greater than ...