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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0answers
12 views

Partial derivatives after a change of variables

Say I have a function of $n$ variables $F(x_{1}, x_{2}, x_{3},...,x_{n})$, where $x_{1} = g_{1}(y_{1}, y_{2}, y_{3},...,y_{m})$, $x_{2} = g_{2}(y_{1}, y_{2}, y_{3},...,y_{m}),\dots, x_{n} = ...
2
votes
1answer
43 views

Verify solution to ODE

I am given the ODE $$\left(f''(x)+\frac{f'(x)}{x} \right) \left(1+f'(x)^2 \right) = f'(x)^2f''(x)$$ and I already know that the solution to this ODE is given by $$f(x)= c \cdot arcosh \left( ...
-1
votes
1answer
57 views

How do I calculate $ \int_{-\infty}^{\infty} e^{-ax^2} \;\mathrm{d}x$ for $a>0 $ [duplicate]

$$ \int_{-\infty}^{\infty} e^{-ax^2} \;\mathrm{d}x \quad\text{for } a>0 $$ I don't even know where to begin to be honest so I haven't made any progress on it. The answer is $\sqrt{\pi/a}$. Is ...
0
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0answers
16 views

solution uniqueness of non-linear Fredholm equations

the equation is $F(x)=G(\int k(x,y)f(y)dy)$ $(*)$ where $f(x)=dF(x)/dx$ is the unknown and it's required to be non-negative. With integral by parts we'll have the form of a non-linear Fredholm ...
1
vote
1answer
23 views

Cartesian to Spherical Coordinates

Can someone help me passing this triple integral from cartesian to spherical coordinates ? Thanks in advance ;) ...
0
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1answer
47 views

Questions about integration

I'm still a bit confused about definite integration although got the basic idea of how to do integration. The problem is to integrate functions on a uniform distribution over [50, 150]. Firstly ...
0
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0answers
22 views

first-order nonlinear ordinary differential equation0

How to solve this differential equation: $$(x^{2}+\ln(y))\cos(2x)+\sin(2x)(xdx+\frac{dy}{2y})=0 $$ I tried to rearrange the equation to the form $\frac{dy}{dx}$ but I couldn't.
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2answers
53 views

integration substitution what am I doing wrong?

integrate $\frac{x^3}{(4+x^2)}$ Let $u = 4+x^2$ so $\frac{du}{2}=xdx$ Then I need to integrate $\frac{(u-4)}{u}$ which comes out as $u-4\ln u$ converting back to $x$, $4+x^2-4 \ln(4+x^2)+C$ But I ...
-2
votes
2answers
67 views

$\int _{ 0 }^{ 1 }{ \frac { { x }^{ t }-1 }{ \ln { x } } dx } $ [duplicate]

How do I solve the following integral: $$\int _{ 0 }^{ 1 }{ \frac { { x }^{ t }-1}{ \ln { x } } dx } $$
3
votes
2answers
62 views

Finding line that divides an area into equal halves.

My question is simple, but I am not getting the answers for some reason. The question is: Consider the area enclosed between the graph of $y = 1 - x^2 $and the $x$ axis. Which line parallel to the ...
0
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0answers
27 views

Volume integral $\int_V y \frac{dy}{dt} + z\frac{dz}{dt} dV$

How do I solve $$\int_V y \frac{dy}{dt} + z\frac{dz}{dt} dV$$ Where $V$ is the volume of a hollow sphere. Usually I would use spherical coordinates, but I don't know how to express $\frac{dy}{dt} $ ...
1
vote
1answer
10 views

Finding the surface integral of a scalar field through an implicitly defined surface

What is the best method of trying to find the surface integral of a scalar field over a certain surface $\Sigma$ that's only defined as being the part of one surface which is cut off by another ...
0
votes
0answers
15 views

Given a set of points, find the plane parallel to plane $p$ where your plane cuts the area in half.

Given a set of point $G=\{(x,y,z) | 0 \le x\le2, 0 \le y \le 2, 0 \le z \le xy\}$ for all $x,y>0$ Find the plane $p$ parallel to plane $zy$ whereas you get two areas equal in size What I did was ...
1
vote
2answers
18 views

Derivation of second moment of area of a circle, a small question

I hope this question will be allowed on math.stackexchange, as the question is a mathematical one even though the subject might be from engineering. I am trying to derive the formula for the second ...
0
votes
1answer
33 views

Taking the partial derivative of an integral

Can I simply take the integral of this function with respect to $t$ by bringing the differential operator under the summation? $$u(x,t)=\int_{-\infty}^{\infty} ...
0
votes
1answer
35 views

Solving the integral $\int_{-\infty}^{\infty} (1+x^2)^{-3/2}$ with $\sinh$, $\cosh$?

I want to solve the following integral: $$\int_{-\infty}^{\infty} (1+x^2)^{-3/2}$$ I thought maybe it's possible with $\sinh$ or $\cosh$ or something similar, but I can't figure it out. Thanks in ...
0
votes
1answer
33 views

Integration of a triple integral

Evaluate the integral $$\int_{z=0}^4\int_{y=0}^1\int_{x=2y}^2{cos(x^2)\over \sqrt{z}}dxdydz$$ by changing the order of integration.
2
votes
3answers
48 views

How do I go about solving the integral of csc x?

So here is a solution provided by another user a couple of years ago and I've seen this solution before but I'm not clear on how or why? What would make me think or tip me off that I should multiply ...
6
votes
2answers
393 views

Integrating over the naturals. What does it mean?

Let $F$ be the power set of $\Bbb{N}$ and consider the measurable space $(\Bbb{N}, F)$. Then what does it mean to take the integral with respect to the measure $\mu(A) = \sum_{a \in A} \frac{1}{a}$. ...
-1
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0answers
29 views

Solving $\frac{\partial}{\partial r} \int_r^x g(y,r) dy = -\frac{\cosh(a\sqrt{x^2-r^2})}{\sqrt{x^2-r^2}}$

I am stuck on this problem. I need to find the correct function, $g(y,r)$, such that $$ \frac{\partial}{\partial r} \int_r^x g(y,r) dy = -\frac{\cosh(a\sqrt{x^2-r^2})}{\sqrt{x^2-r^2}} $$ So far I am ...
2
votes
4answers
145 views

How to evaluate this indefinite integration $\int \frac{\tan^4 \theta d \theta}{1-\tan^2 \theta}$?

I have to solve this indefinite integration $$\int \frac{\tan^4 \theta d \theta}{1-\tan^2 \theta}$$ I tried it as follows $$I=\int\frac{(\sec^2 \theta-1)\tan^2 \theta d \theta}{1-\tan^2 ...
0
votes
6answers
87 views

integrate $1/(x(x^2-1)^{1/2})$

$$\int\frac{1}{x (x^2-1)^{1/2}} \, dx=\text{ ?}$$ 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 ...
3
votes
3answers
87 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 ...
7
votes
2answers
138 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
103 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
35 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
50 views

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

Help with this integral, please! $$\int_{0}^{1} \frac{x^3}{\sqrt{1+x^4}}dx$$
6
votes
1answer
126 views

Integration of $\frac{e^{\cos^2x}+\ln(1+x)}{10^{x^3}\arctan(\sqrt{x})}$, possibly numerical

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
40 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
59 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 ...
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2answers
30 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
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0answers
15 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
119 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
64 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
37 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
22 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
89 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
26 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
125 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
49 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
108 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
18 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.
3
votes
0answers
40 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
187 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
43 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 ...