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 ...

learn more… | top users | synonyms (2)

5
votes
1answer
56 views

Integral: $\int_0^{\pi} \frac{x}{x^2+\ln^2(2\sin x)}\,dx$

I am trying to solve the following by elementary methods: $$\int_0^{\pi} \frac{x}{x^2+\ln^2(2\sin x)}\,dx$$ I wrote the integral as: $$\Re\int_0^{\pi} \frac{dx}{x-i\ln(2\sin x)}$$ But I don't find ...
0
votes
1answer
11 views

Proving some properties about the expected first order statistic (maximum) with respect to sample size.

Question: Consider $n$ random variables $x_1, x_2,\cdots x_n\sim \mathcal{N}(0,1)$. The expected value of the $i$th order statistic (the maximum) can be written as ...
0
votes
0answers
46 views

Evaluate $\int\left({\frac{\arctan x}{\arctan x-x}}\right)^2 \,dx$ [duplicate]

As the title shown, how to evaluate the indefinite integral $$\int\left({\frac{\arctan x}{\arctan x-x}}\right)^2 \,dx\ ?$$ Thanks.
4
votes
0answers
59 views

Prove ${\large\int}_0^\infty\left({_2F_1}\left(\frac16,\frac12;\frac13;-x\right)\right)^{12}dx\stackrel{\color{#808080}?}=\frac{80663}{153090}$

I discovered the following conjectured identity numerically (it holds with at least $1000$ digits of precision). How can I prove it? ...
0
votes
2answers
38 views

Volume of trig function around y-axis

I have this question and it's the first kind of question I'm doing involving finding volume so I just would like some help solving this question: Find the volume created by revolving the curve $ \ ...
4
votes
3answers
108 views

Evaluate $\int_0^1\frac{x^a-x^{-a}}{x-1}dx$

I have heard that: $$\int_0^1\frac{x^a-x^{-a}}{x-1}dx=\frac1 a-\pi\cot(\pi a)$$ when $-1<a<1$. How would I prove this? That doesn't have an elementary indefinite integral, but the definite ...
9
votes
0answers
101 views

Evaluating $\int_0^\pi\arctan\left(\frac{\log\sin x}{x}\right)\mathrm{d}x$

I found the following integral as a by product of another one. It has a nice closed form. $$ \displaystyle \int_{0}^{\pi} \arctan \left( \frac{\log \sin x}{x} \right) \mathrm{d}x $$ ...
1
vote
1answer
38 views

Area of solid revolution using integration.

When we calculate the volume of a solid generated by rotating a curve around $x$-axis, We divide it into disks. So ,we get $dv = \pi r^2 dx$. where $r=y$ and then we integrate. That OK, but when ...
0
votes
1answer
16 views

Average value for multiple integrals

If there is a function $f(x,y)$ and we want to find the average value over a region $R$ defined by $0<x<1$ and $0<y<x$, how is that computed? I know that it would be something like this: ...
3
votes
1answer
36 views

Generalising integration by parts for the product of more than two functions

Just as the product rule can be generalised to the product of more than two functions, i.e. $$\frac{d}{dx} \left [ \prod_{i=1}^k f_i(x) \right ] = \sum_{i=1}^k \left(\frac{d}{dx} f_i(x) \prod_{j\ne ...
3
votes
2answers
72 views

Intuition about Taking an Integral

My hope is to personally develop some further intuition for taking an integral (measuring the area under a curve). Consider a normal distribution and I need the area under the curve from $a$ to $b$. I ...
0
votes
0answers
54 views

Is the following integration of possible?

How to solve the following problem? $\int x^4/(1-x^4)^{3⁄2}dx$ I have tried the substitution, x=sinz, but failed.
0
votes
0answers
15 views

Volume of Solid of Revolution

This problem is giving me some trouble: The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by any method. x = (y − 5)^2, x = 4; ...
1
vote
0answers
33 views

Integrating $\int_0^1 dx\,\ln(x-a)/(x-b)$ paying attention to cuts.

I am trying to compute the following integral, for complex $a$ and $b$ $$\int ^1 _0 dx \frac{\ln(x-a)}{x-b}$$ by turning it into something in terms of dilogarithms. But for certain values of $a$ ...
0
votes
1answer
32 views

Use triple integrals to integrate over a tetrahedron

Integrate $f(x, y, z) = x^2 + y^2 - z$ over the tetrahedron with vertices $(0, 0, 0), (1, 1, 0), (0, 1, 0), (0, 0, 3)$. I need to use triple integrals to solve this, so I made a diagram and set the ...
0
votes
1answer
54 views

Integral of the exponential function

I am searching the indefinite integral of this function: $\dfrac{\exp(x)}{(1+x)^{5/3}}$. Thank you alot.
2
votes
2answers
44 views

Duo Fresnel-like integrals $(??)$

I really wonder how I can prove the following integrals. $$\int_0^\infty \sin ax^2\cos 2bx\, dx=\frac{1}{2}\sqrt{\frac{\pi}{2a}}\left(\cos \frac{b^2}{a}-\sin\frac{b^2}{a}\right)$$ and ...
0
votes
1answer
57 views

Explain why $\big(\int_{-\infty}^{\infty}e^{-z^2/2}dz \big)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(z^2 + u^2)/2}dzdu$

I came across the following when studying a proof related to the normal distribution: $$\left(\int_{-\infty}^{\infty}e^{-z^2/2}\ dz \right)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(z^2 ...
4
votes
0answers
33 views

How to integrate scalar field over quarter torus? Infinite series does not converge.

This seems to be physics question, but the problem just concerns math. Preface If one wants to calculate the permeance $P$ of a rectangular bar: it is an easy task: $$P = \frac{\mu a b}{L} ...
0
votes
1answer
10 views

Reference for transformation of integrals over Lipschitz boundaries

Let $D\subseteq\mathbb{R}^d$, $d\ge 2$, be a bounded Lipschitz domain. Then according to page 314 of [Function spaces, Alois Kufner, 1977] one can define a surface integral of a real valued function ...
1
vote
1answer
65 views

Seemingly hard integrals which are made easy via differentiation under the integral sign a.k.a Feynman Integration

I recently discovered Differentiation under the integral sign a.k.a Feynman Integration and I read an article which says it can be substituted for contour integration. Therefore, I am assuming this ...
4
votes
1answer
47 views

$\int_\mathbb{R} \bigg( \frac{1}{h} \int_x^{x+h} |f(t)| dt\bigg) dx= ||f||_{L^1}$?

$$\int_\mathbb{R} \bigg( \frac{1}{h} \int_x^{x+h} |f(t)| dt\bigg) dx= ||f||_{L^1} \;\;?$$ I worked out that the equality holds for each $\chi_{[a,b]}$, therefore it holds for each piecewise ...
1
vote
1answer
47 views

Definite integral involving powers and logarithm

Does the following definite integral have a closed form solution? $$ \int_0^1 x^a(1-x^b)^c\ln(1-x)\ dx $$
0
votes
5answers
74 views

Proving integrals of $f(\sin(x))$ and $f(\cos(x))$ are equal

Prove that for every continuous function $f$, $\displaystyle\int_{0}^{\pi/2} f(\sin(x))dx = \int_{0}^{\pi/2} f(\cos(x))dx $. I am not really sure how to tackle this, but nevertheless here is my ...
0
votes
1answer
54 views

Finite integral with goniometric functions, $\int_0^{\infty} \frac{8\sin^4(\pi f t)\tan^2(\pi f/2)}{(\pi^4 \tau^2 f^3) }df$

I have difficulties trying to find an algebraic solutions of the following integral: The $\tau$ in this formula is an integer, which is a very important fact because only then this integral is ...
3
votes
2answers
72 views

Evaluate $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$

I need to evaluate the following integral: $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$$ I thought of evaluating the iterated integral ...
1
vote
4answers
87 views

Antiderivative of $\frac{1}{1+\sin {x} +\cos {x}}$

How do we arrive at the following integral $$\displaystyle\int\dfrac{dx}{1+\sin {x}+\cos {x}}=\log {\left(\sin {\frac{x}{2}}+\cos {\frac{x}{2}}\right)}-\log {\left(\cos {\frac{x}{2}}\right)}+C\ ?$$
8
votes
2answers
73 views

Evaluating $\int_{-\infty}^\infty \frac{\sin x}{x-i} dx$

I would like to evaluate the integral $$\int_{-\infty}^\infty \frac{\sin x}{x-i} dx,$$ which I believe should be equal to $\frac{\pi}{e}$. However, I cannot reproduce this result by hand. My work is ...
1
vote
0answers
38 views

Taking derivative under the integral sign

Reading a textbook and stuck on this one detail... would like to confirm my understanding. The book defines a function $\eta \in C^1(\mathbb{R})$ satisfying $0 \leq \eta \leq 1$, $0 \leq \eta^\prime ...
2
votes
1answer
53 views

What does this paper mean by “$f(x)$ is practically a rational function”?

The paper "Infinite integrals involving Bessel functions by contour integration" by Qiong-Gui Lin gives a method to solve integrals of the form $\intop_{0}^{\infty}x^{v}f(x)J_{v}(qx)\, dx$. One of the ...
3
votes
1answer
44 views

Double integral proof, where is my mistake?

The bounds are 0 < x < b , 0 < y < b. $$ \int_0^b \int_0^b e^{-(x^{2}+y^{2})} dxdy $$ Since it is a square, x=y so we can write: = $$ (\int_0^b e^{-(x^{2}+x^{2})} )^{2} dxdy $$ = $$ ...
3
votes
3answers
256 views

Double integral proofs

Show that $$\renewcommand{\intd}{\,\mathrm{d}} \iint_{D(R)} e^{-(x^2+y^2)} \intd x \intd y = \pi \left(1 - e^{-R^2}\right)$$ where $D(R)$ is the disc of radius $R$ with center $(0,0).$ I ...
3
votes
0answers
81 views

Integral ${\large\int}_0^\infty\frac{\ln x}{1+x}\sqrt{\frac{x+\sqrt{1+x^2}}{1+x^2}}\ dx$

Please help me to evaluate this integral: $$I={\large\int}_0^\infty\frac{\ln x}{1+x}\sqrt{\frac{x+\sqrt{1+x^2}}{1+x^2}}\ dx.\tag1$$ Mathematica could not evaluate it in a closed form. A numerical ...
2
votes
2answers
22 views

Confusion about Spherical Coordinates Transformation

We have a function $$f(x,y,z) = \frac{e^{-x^2 -y^2 -z^2}}{\sqrt{x^2+y^2+z^2}}$$ and we want to integrate it over the whole $\mathbb{R}^3$. Then what i got is the following: $$\int_{\mathbb{R}^3}^ \! ...
0
votes
1answer
55 views

Just calculus, the integral = 0 and the argument inside integral = 0?

It is really hard for me to make a title to describe my question. Below is my question: Suppose $f(y-x)$ is a known Gaussian function defined as $$ f(y-x) = \frac{1}{\sqrt{2\pi}} \exp ...
1
vote
1answer
27 views

Finite Measure Space: Integral Closure = Bochner Integral

I can't sleep for so long time as the integral gives me headaches. I was looking for so many approaches. Now another one. Hope this works... Let $\Omega$ be a finite measure space and $E$ a Banach ...
2
votes
1answer
44 views

Properties of the Double Layer Potential

Consider the double layer potential $$ W_{\nu}(x) = \int_{\partial\Omega} \nu(y) \frac{\partial}{\partial n_y}\left( \frac{1}{|x - y|} \right) d \sigma_y $$ for a bounded region $\Omega \in \mathbb ...
8
votes
1answer
140 views

Evaluate $\int_{0}^{\large\frac{\pi}{4}} \ln {(\sin x)}\cdot\ln {(\cos x)} \left(\frac{\ln{(\sin x)}}{\cot x}+\frac{\ln {(\cos x)}}{\tan x}\right)dx$

How do I find the value of this integral? $$I=\int_{0}^{\Large\frac{\pi}{4}} \ln {(\sin x)}\cdot\ln {(\cos x)} \left(\dfrac{\ln{(\sin x)}}{\cot x}+\dfrac{\ln {(\cos x)}}{\tan x}\right)dx$$ I tried ...
3
votes
1answer
81 views

any simple method to do integration?

$$\int_{-2}^{x^{2}-2x}e^{t}.e^{t^2} dt = ?$$ What i did is... on rewriting it , $$\int_{-2}^{x^{2}-2x}e^{t+t^2} dt=\frac{e^{t+t^2}}{t^2/2+t^3/3} $$ and then substituting limits is very long process ...
1
vote
0answers
14 views

existence and uniqueness of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
6
votes
1answer
145 views

An equivalent for $\int_0^1\left(\frac{1}{\log x}+\frac{1}{1-x}\right)^n\;dx$

Set $$ I_n :=\int_0^1\left(\frac{1}{\log x} + \frac{1}{1-x}\right)^n \:\mathrm{d}x \qquad n=1,2,3,.... $$ We have $$I_1 =\gamma, \quad I_2 =\log (2 \pi) - \frac 32, \quad I_3 = 6 \log A - ...
0
votes
1answer
25 views

Set function integal

We have a vector $y$ ($\sum_i y_i=1$). Define $S(r) = \{i, y_i\geq r \}$. Here is an integral $\int_{0}^{\infty} |S(r)| dr=\sum_i y_i$. I don't know why the integral is correct. Can anybody help me?
1
vote
1answer
68 views

An integral representation for $\psi$

Let $\displaystyle \gamma$ denote the Euler constant defined by $\displaystyle \gamma := \lim\limits_{n \to \infty} \left(\frac11+\frac12+\cdots+\frac1n- \log n\right)$. Here is an integral for ...
0
votes
1answer
20 views

Calculating the area of a region using a mapping

The region: $\{{(x,y) \mid x^{2} < y < 2x^{2}, 2y^{2}<x<3y^{2}, x > 0, y > 0}\}$ The mapping: $u = y/x^{2}$, $v = x/y^{2}$ I calculated the jacobian to be $\frac 34$ which means ...
2
votes
1answer
65 views

Evaluate the area of the region bounded by the ellipse, where is my mistake?

$ (10x^2+6xy+y^2=2)$ => $ ((x/\sqrt2)^{2} + ((3x+y)/\sqrt2))^{2} = 1 $ so if I change the variables to $u$ and $v$, $u = x/\sqrt2$ $v= (3x+y)/\sqrt2) $ Then my bounds of integration become $-1 ...
1
vote
0answers
27 views

Not lebesgue integrable function?

I want to consider the function $f:[-1,1]\times [-1,1]\rightarrow \mathbb R:f(x,y)= \begin{cases} \frac{xy}{(x^2+y^2)^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases} $ And I have ...
0
votes
0answers
16 views

How do you obtain the version of Simpson's rule required as well as deduce the composite integration rule? [on hold]

Consider the function $$g(x)=f(a+(x−1)h)$$ and obtain a version of Simpson’s rule applicable to an integral $$\int_{a+h}^{a−h}f(x)dx.$$ Then deduce the composite integration rule ...
2
votes
2answers
88 views

Evaluation of $\int\frac{\sqrt{\cos 2x}}{\sin x}dx$

Evaluation of $$\displaystyle \int\frac{\sqrt{\cos 2x}}{\sin x}dx$$ $\bf{My\; Try::}$ Let $\displaystyle I = \int\frac{\sqrt{\cos 2x}}{\sin x}dx = \int\frac{\cos 2x}{\sin^2 x\sqrt{\cos 2x}}\sin xdx ...
15
votes
1answer
199 views

How find this integral $\int_{0}^{\infty}\frac{dx}{(1+x^2)(1+r^2x^2)(1+r^4x^2)(1+r^6x^2)\cdots}$

prove that this integral $$\int_{0}^{\infty}\dfrac{dx}{(1+x^2)(1+r^2x^2)(1+r^4x^2)(1+r^6x^2)\cdots}= \dfrac{\pi}{2(1+r+r^3+r^6+r^{10}+\cdots}$$ for this integral,I can't find it.and I don't know how ...
0
votes
1answer
34 views

How do you solve the second part of the question where i am required to derive Simpson’s integration rule?

When $v(x) = A + Bx + Cx(x − 1)$ show that $$\int_0^2v(x)dx= 2A + 2B + \frac23.$$ By choosing A,B and C so that $y = v(x)$ fits a given curve $y = g(x)$ at $x = 0$, $x = 1$ and $x = 2$ derive ...