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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1answer
52 views

Find the value of the integral on the contour C

Ok, so I'm trying to figure out this problem. It asks to find the value of the contour integral $\dfrac{e^z}{z^2(z-\pi i)}$ on the contour $C$ shown in the following figure I believe that in order ...
1
vote
0answers
84 views

Evaluate the triple integral bounded by a cylinder and planes

Evaluate the triple integral $\displaystyle\iiint\limits_{W}z\,dx\,dy\,dz$ over the region $W$ which is bounded by the planes $x=0$, $y=0$, $z=0$, $z=1$, and the cylinder $x^2+y^2=1$ in the first ...
10
votes
2answers
158 views

Closed form of $\displaystyle\int_{0}^{\pi/4}\int_{\pi/2}^{\pi}\frac{(\cos x-\sin x)^{y-2}}{(\cos x+\sin x)^{y+2}}\, dy\, dx$

Can the following double integral be evaluated analytically \begin{equation} I=\int_{0}^{\Large\frac{\pi}{4}}\int_{\Large\frac{\pi}{2}}^{\large\pi}\frac{(\cos x-\sin x)^{y-2}}{(\cos x+\sin ...
0
votes
1answer
43 views

Writing iterated integral of a function

Write an iterated integral of a function f for the region given by a triangle with vertices at point (1,1), (1,2), (3,0). I figured that I'm supposed to first find the equations of the three lines ...
0
votes
1answer
26 views

Writing an iterated integral of a function

Write an iterated integral of a function $f$ for the region $ x \ge 0, y \ge 0,$ between sinusoid $y = \sin x$ and line $y =\dfrac{2}{\pi}x$ I am stuck when trying to find the boundaries for my ...
0
votes
1answer
50 views

Evaluate the triple integral of $x^2+y^2$ where D is a pyramid

Evaluate the triple integral of $f(x,y) = x^2+y^2$ where the region of integration is the pyramid with top vertex at $(0,0,1)$ and base vertices at $(0,0,0)$, $(1,0,0)$, $(0,1,0)$, and $(1,1,0)$. I ...
0
votes
1answer
55 views

Double Integral over region

Problem: Compute the double integral $$\iint_R xsin(y^2) dA$$ over the region $$R { (x,y): 0≤x≤3 , x^2≤y≤9 } $$ Determine in which order it would be possible to perform the integration and ...
5
votes
1answer
127 views

Show $\int_0^\infty f\left(x+\frac{1}{x}\right)\,\frac{\ln x}{x}\,dx=0$ if $f(x)$ is a bounded non-negative function

Lemma If $f(x)$ is a bounded non-negative function, then \begin{equation}\int_0^\infty f\left(x+\frac{1}{x}\right)\,\frac{\ln x}{x}\,dx=0\end{equation} I found this lemma on internet and ...
1
vote
2answers
129 views

Find volume above cone within sphere

My objective: Using spherical coordinates, set up and compute an integral to find the volume of the ice-cream-cone shaped solid lying above the cone $z = \sqrt{x^2 + y^2}$ and below the sphere ...
0
votes
1answer
30 views

Understanding the following expression in an integration

I cannot understand how the parts of this proof circled in red are obtained.
8
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3answers
298 views

Prove $\int_0^1\frac{\ln2-\ln\left(1+x^2\right)}{1-x}\,dx=\frac{5\pi^2}{48}-\frac{\ln^22}{4}$

How does one prove the following integral \begin{equation} \int_0^1\frac{\ln2-\ln\left(1+x^2\right)}{1-x}\,dx=\frac{5\pi^2}{48}-\frac{\ln^22}{4} \end{equation} Wolfram Alpha and Mathematica can ...
0
votes
1answer
55 views

Properties of continuity

Let $f,g :[a,b]\to\mathbb{R}$ be continuous functions such that $$\int\limits_c^df(x)\leq \int\limits_c^dg(x)dx$$ whenever a$\leq$c$<$d$\leq$b. I need to show that $f(x)\leq g(x)$. I have the ...
0
votes
0answers
21 views

Integration matrix

I want to do integration(summation) of a signal(x) using matrix multiplication. I am looking for a transformation matrix, I corresponding to integration such that F = I * x , where x is the signal ...
9
votes
2answers
107 views

A question on cosine integral

So I've read a book and found myself stumped in this integral: $$\int_{0}^{\pi} \frac{\cos(n\theta)}{b^2-a^2\cos(2\theta)}\, d\theta=\begin{cases} \,\,0 &,\quad\mbox{if}\,\, n\,\,\mbox{is ...
0
votes
2answers
37 views

Using solids of revolution to find the volume of a sphere cap

I want to find the volume of a sphere cap using the solids of revolution method. Let the sphere have radius $r$ and the cap have height $h$. Then the volume of the cap is given by $\pi ...
1
vote
1answer
61 views

How to integrate $\int_{-1}^1\frac{1}{a + bx }dx$, where $a,b\in \mathbb{C}$ without using branch cuts.

Is there a way to integrate $$\int_{-1}^1\frac{1}{a + bx }dx,\,\,\,\,(*) $$ where $a,b\in \mathbb{C}$ without using branch-cuts? I was approached with such an integral relatively early in my text, and ...
3
votes
1answer
131 views

Closed form: $\int_0^\pi \left( \frac{2 + 2\cos(x) -\cos((k-\frac{1}{2})x) -2\cos((k+\frac{1}{2})x) - cos((k+\frac{3}{2})x)}{1-\cos(2x)}\right)dx $

Find a closed form for the following definite integral: $$ I =\int_0^\pi \left( \frac{2 + 2\cos (x) - \cos((k-\frac{1}{2})x) - 2\cos ((k+\frac{1}{2})x) - \cos((k+\frac{3}{2})x)}{1-\cos(2x)}\right) ...
7
votes
1answer
92 views

Finding a double integral $\int_1^\infty\int_0^\infty\frac{1}{(x^3+y^3)^3}\mathrm{d}x\ \mathrm{d}y=\frac{10\pi}{189\sqrt3}$

How do we prove that $$\int_1^\infty\int_0^\infty\dfrac{1}{(x^3+y^3)^3}\mathrm{d}x\ \mathrm{d}y=\dfrac{10\pi}{189\sqrt3}$$ I tried to expand and use partial fraction, but in vain. I don't have a clue ...
1
vote
1answer
30 views

How to compute the integral of that area?

The area is given by $ 0 \le x+y \le 4-(x-y)^2 $ ? By magic the inequalities were transformed into $ 0 \le \sqrt 2 \ u \le 4 - 2v^2 $ and after that computing the integral became almost trivial. I ...
1
vote
1answer
37 views

Manipulation before integrating by parts

If it is given that$$\int_0^4e^{(x-2)^4}dx=A $$ Then find the value of $$ \int_0^4xe^{(x-2)^4}dx$$ How do we manipulate the integral so that integration by parts can be used thereafter? Letting $x$ ...
2
votes
2answers
75 views

How to find a non-Gaussian function f(x) that satisfies the following condition:

$\lim_{X \to \infty} \int_0^Xf(x)^2 > 2(\lim_{X \to \infty}\int_0^Xf(x))^2$
2
votes
1answer
49 views

Solution of Matrix ODE

Specifications It is given that $ \psi'(s)=(A+Bs)\psi(s)\tag 1$ where A,B are constant $3 \times 3$ skew symmetric matrices with determinant $0$ $\psi(s)$ has determinant $1$ , orthogonal and has ...
6
votes
1answer
121 views

I am having trouble with this integral from the 2012 MIT Integration Bee

$$\int\frac{dx}{(1+\sqrt{x})\sqrt{x-x^2}} $$ Could someone explain to me how to integrate this integral. Thank you and cheers.
19
votes
4answers
335 views

An integral by O. Furdui $\int_0^1 \log^2(\sqrt{1+x}-\sqrt{1-x}) \ dx$

The following integral was proposed in a paper by O. Furdui, namely $$\int_0^1 \log^2(\sqrt{1+x}-\sqrt{1-x}) \ dx$$ and then the generalization $$\int_0^1 \log^2(\sqrt[k]{1+x}-\sqrt[k]{1-x}) \ ...
0
votes
0answers
20 views

Integral of $\int_{0}^{z^*}0dz$

So I have to find the max upward velocity for an air parcel. Anyways so $U$ is the vertical velocity and $z^*$ is the height were the max vertical velocity occurs. From this I got the integral of ...
1
vote
1answer
50 views

Am I doing something wrong with this improper integral?

I have a little discussion with my friends about my "resolution" and calculation of $$\int_{-\infty}^1 e^{4x} \, dx.$$ I did $$\int_{-\infty}^1 e^{4x} \, dx =\int_{-1}^{\infty} e^{-4x} \, dx = ...
9
votes
2answers
119 views

How to prove $\int_{0}^{-1} \frac{\operatorname{Li}_2(x)}{(1-x)^2} dx=\frac{\pi^2}{24}-\frac{\ln^2(2)}{2} $

$\def\Li{\operatorname{Li}}$ I wonder how to prove: $$ \int_{0}^{-1} \frac{\Li_2(x)}{(1-x)^2} dx=\frac{\pi^2}{24}-\frac{\ln^2(2)}{2} $$ I'm not used to polylogarithm, so I don't know how to tackle it. ...
1
vote
0answers
60 views

Computing an integral explicitly

I have the following integral: let $k$ be a positive fixed integer and $\varepsilon \in (0,1/k)$. Let $t>0$ and $r>0$. We consider $$\int_0^1 \int_0^{t/2} (t^{2\varepsilon} - ...
2
votes
1answer
63 views

How to approximate this nasty exponential function with an integral?

What is the best way to approximate a function of the following form, $$ \text{exp}\left(-\int_{y}^{+\infty} f(x)\ dx \right)$$ Any approximation to this, does taylor series work? The reason I am ...
1
vote
3answers
102 views

Integrate $ \int_0^{\infty} \! x^2 e^{-ax^2} \, \mathrm{d}x $ [closed]

Integrate $$ \int_0^{\infty} \! x^2 e^{-ax^2} \, \mathrm{d}x $$ We may assume without proof: $$ \int_0^{\infty} \! e^{-x^2} \, \mathrm{d}x = \frac{\sqrt{\pi}}{2}$$
1
vote
0answers
27 views

The inverse of the integration over a ball with radius $\epsilon$

First of all sorry for the nondescript title, but this seemed like the most suitable one. Now let $d\geq2, D\subset \mathbb{R}$ a domain and $G:D\times D\rightarrow[0,\infty]$ continuous. Define ...
1
vote
3answers
301 views

integral calculation is wrong. Why?

$$\int \sqrt{1-x^2} dx = \int \sqrt{1-\sin^2t} \cdot dt= \int \sqrt {\cos^2 t} \cdot dt= \int \cos t \cdot dt = \sin t +C = x +C$$ The answer is wrong. Why?
2
votes
1answer
97 views

Changing the limits of integration when function is infinite

Suppose I have an integral: $$\int_0^{\pi/2}f(b\tan x)dx$$ With $b$ being some positive parameter. Now if I want to change variables in this way: $b\tan x = \tan t$, how will the upper limit change? ...
5
votes
1answer
94 views

Integrating over a power of the infinitesimal

I don't know if the title makes sense (or if the question makes sense at all for that matter) but here I go. Suppose I have a piecewise constant function $y=f(x)$ with $x,y\in\mathbb{R}^+$, described ...
1
vote
0answers
68 views

Expand $\int_{-1}^0 e^{a\cos{\theta}}J_0(b\sin{\theta})\,d\cos{\theta}$ in spherical harmonics.

I want to solve the integral (a probability density function) $$ g(\gamma)=\int_{-1}^0 e^{-f\cos{\theta}\cos{\gamma}}J_0(-if\sin{\theta}\sin{\gamma})\,d\cos{\theta} $$ numerically, everything is ...
2
votes
4answers
111 views

Evaluate $\int_0^2\frac{x^5}{\sqrt{x^3+6}}\,dx.$

I am stuck on the following integral: $\displaystyle\int_0^2\dfrac{x^5}{\sqrt{x^3+6}}\,dx.$ I have no idea how one can work it out. Normally I'd try $u=x^3+6$ but this surely does not work here.
1
vote
1answer
82 views

Finding a Counter Example - Limits of integrals of an increasing sequence of Borel measurable functions

I need to find a counter example to the following problem. I'm trying to think of some, but maybe I'm not creative. I'm not sure. Let $h$ and $h_1, h_2, h_3, ...$ be Borel measurable functions such ...
8
votes
2answers
142 views

Finding the closed form of $\int_0^1 \frac{(1-x+x\log x)\operatorname{Li}_3(x)}{x(x-1) \log x} \ dx$

$\def\Li{{\rm{Li}}}$Here I have a question I just received, and still trying to find a proper starting point $$\int_0^1 \frac{(1-x+x\log x)\Li_3(x)}{x(x-1) \log x} \ dx$$ What starting point would ...
2
votes
1answer
103 views

how to integrate $\mathrm{arcsin}\left(x^{15}\right)$?

Integral by parts: $$ I = x\sin^{-1}\left(x^{15}\right) - \int\frac{15x^{15}}{\sqrt{1-x^{30}}}dx $$ then what? The answer by wolfram gives an answer contains hypergeometric ${}_2F_1$ function,because ...
1
vote
1answer
29 views

Prove the f is integrable when $f(x)=(1-x^4)^\frac{1}{2}$

Let $f(x)= (1-x^4)^{1/2}$ by $f:[0,1]\to \mathbb{R}$. I need to prove the $f$ is integrable on $[0,1]$. I think I need to use a partition but I have no idea how to prove the integrability or where to ...
0
votes
0answers
47 views

A question related to “supremum”

Let $f\in L_{p}([0,1])$ and $[a,c]\subset [0,1].$ Is there an inequality between the following quantities? $$ \underset{|h|\leq b-a}{\sup}\int_{a}^{b}|f(x+h)-f(x)|^{p}dx+ \underset{|h|\leq ...
10
votes
2answers
153 views

Evaluation of a tough double integral

This is an integral coming from personal research, and very important to me, but it does not seem an easy job to do. If a solution is not possible then I'd be glad with a closed form only. ...
0
votes
1answer
33 views

Am I correct with this change of variable?

I have been solving a problem from a paper I read related to poisson point processes and for some reason I am not reaching the same result the paper has. The problem is re-expressing an expression by ...
0
votes
1answer
46 views

Find the volume of the solid generated by revolving the region bounded by $y=x$ and $y=x^2$ about the line $y=x$

Find the volume of the solid generated by revolving the region bounded by $y=x$ and $y=x^2$ about the line $y=x$ I am confused, how do we approach such problems, where the rotation lines are not ...
0
votes
1answer
339 views

Splitting Up Integrals and Multiplying Them

$$I_x = \int_0^b\int_0^h\rho y^2\,\mathrm{d}y\mathrm{d}x$$ So here's the current problem I'm working on, just for an example. I saw my teacher break up a triple integral in class today then multiply ...
2
votes
1answer
146 views

Nasty integration?

So I am trying to solve the following integral and apparently its not integrable or I might be wrong. Not even computer software can integrate. Can anyone tell me if this is integrable or not? The ...
1
vote
0answers
18 views

Integrate Two Dot-Product-Power Terms

I need to compute something like the following integral:$$ \int_{\Omega_\vec{a}} \left< \vec{x} \cdot \vec{b} \right>^n \left(\vec{x}\cdot\vec{a}\right)^m d\vec{x} $$Notational issues: The ...
0
votes
1answer
39 views

Radon transforms and determining a separable function

I am interested in the Radon transformation for separable functions $F(x,y) = f(x)g(y)$. Why is it in tomography that a separable function is determined completely by two of its projections ? And ...
1
vote
2answers
53 views

Why is $F(x)$ continuous at $x=0$?

Let $$f(x) = \left\{ \begin{array}{ll} x & \mbox{if } x < 0 \\ \sin x & \mbox{if } x \ge 0 \end{array} \right.$$ $F(x)$ the anti-derivative should be $\frac{x^2}{2} + C_1$ for ...
1
vote
4answers
75 views

Derivative of $\int_{x}^{0} \frac{\cos xt}{t} dt$

I am working on the following problem: Find the derivative of $f(x)=\displaystyle \int_{x}^{0}\displaystyle\frac{\cos xt}{t}dt$. The answer I am supposed to get is $\displaystyle ...