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 ...

learn more… | top users | synonyms (3)

0
votes
0answers
6 views

Finding the mean of $x \mapsto |x|^p$ on the unit ball in $\mathbb{R}^n$ without calculating the volume of the unit ball.

Find the mean of the function $x\mapsto |x|^p$ on the ball $\{x:|x|<1\}\subset\mathbb{R}^n$ for $p\in(0, \infty)$. (Hint: You do not need the volume of the ball). I tried doing something ...
1
vote
1answer
20 views

Showing that $2\int_0^\infty {\cosh(x)-1\over x(e^{ax}-1)}dx=\ln\left({\pi \over a\sin\left({\pi\over a }\right)}\right)$

$$2\int_{0}^{\infty}{\cosh(x)-1\over x(e^{ax}-1)}dx=\ln\left({\pi \over a\sin\left({\pi\over a }\right)}\right)$$ $$\cos(x)=1+{x^2\over 2!}+{x^4\over 4!}+{x^6\over 6!}+\cdots$$ $${\cos(x)-1\over ...
2
votes
1answer
45 views

Find the volume of the region enclosed by $x^2+y^2+z^2=2$ and $x^2+y^2=z$.

Find the volume of the region enclosed by $x^2+y^2+z^2=2$ and $x^2+y^2=z$. I tried to solve the problem above by doing a change of variables to the spherical coordinate system, that is, $x= \rho ...
0
votes
0answers
8 views

FTC examples, questions

Evaluate the following using fundamental theorem of calculus. A) $$F(x)={d\over dx }\int_2^{e^x}\ \frac{\log z}{z} dz$$ B) $$f'(x)=\log_{e^x} e^{\int_0^{x}\ e^{u^2} du}$$ I tried for A) use ...
0
votes
1answer
18 views

Can quaternions be used to represent rotation rate?

A quaternion is a useful tool for representing a rotation, or change in attitude. If a quaternion $q$ represents a rotation, and $v$ a vector, then $v'=qvq^*$ rotates the vector, where the multiply ...
-1
votes
1answer
19 views

What is the most practical way to express in mathematical notation the indefinite integral of the gamma function?

I checked on WolframAlpha (I am not good at calculating integrals) and it said there are no elementary functions to express it and gave me a horribly complex (I think... I didn't examine it very ...
2
votes
1answer
34 views

Prove that $\int \frac{dx}{x (\ln x)^{k+1}} = \frac{1}{(1-k)(\ln x)^{k}}$ $k \in \mathbb{R}$

For fun I studied the following integral $$ \int \frac{dx}{x (\ln x)^{k+1}} $$ and of course I wanted to solve it. I started with $k=1$, which gives us the following $$ \int \frac{dx}{x (\ln x)^2} ...
21
votes
2answers
358 views

Evaluating :$\int \frac{1}{x^{10} + x}dx$

$$\int \frac{1}{x^{10} + x}dx$$ My solution : $$\begin{align*} \int\frac{1}{x^{10}+x}\,dx&=\int\left(\frac{x^9+1}{x^{10}+x}-\frac{x^9}{x^{10}+x}\right)\,dx\\ ...
4
votes
1answer
7k views

Work to pump water from a cylindrical tank

We have a cylindrical tank with radius 2m and length 10m filled with water. How much work does it tank to pump the water out of the tank from the top? My attempt at the problem goes as follow. $g$ ...
3
votes
0answers
96 views

Evaluate$ \iint_S\frac{x+y+z}{(x^2+y^2+z^2)^{3/2}}ds$

Evaluate $\displaystyle \iint_S\frac{x+y+z}{(x^2+y^2+z^2)^{3/2}}ds$ where $S$ is the unit disk centered at $(1,1,1)$ and lies on the plane $x+y+z=3$. So my inegral becomes $\displaystyle ...
1
vote
0answers
32 views

Integral solution of separable differential equation

On page 524 of Tenenbaum's Introduction to Analytic and Probabilistic Number Theory (3rd edition) it is essentially stated that the solution to the first-order differential equation $$y' = e^{-x}y/x ...
2
votes
0answers
23 views

Solving the integral $\int_{c}^{\frac{x}{i}+c}\int_{(i-1)z}^{x+ic-z}[x-(i-1)z]^{i-2}e^{-x-z}\,\mathrm dx\,\mathrm dz$

I have following integral to solve $$\int_{c}^{\frac{x}{i}+c}\int_{(i-1)z}^{x+ic-z}[x-(i-1)z]^{i-2}e^{-x-z} \,\mathrm dx \,\mathrm dz$$ where $i$ is positive integer and $c>0$. After doing some ...
1
vote
1answer
20 views

Derivative of convolution type integral equation with respect to time $ x(t) = \int\limits_{0}^t \gamma^{t-s} u(s)ds$

Consider the equation: $$x(t) = \int\limits_{0}^t \gamma^{t-s} u(s)ds$$ where $x, u$ are functions and $\gamma \in \mathbb{R}_{>0}$ I am trying to obtain the derivative of this equation with ...
6
votes
2answers
102 views

Prove $\int_{0}^{\infty}\frac{4x^2}{(x^4+2x^2+2)^2}dx=\frac{\pi}{4}\sqrt{5\sqrt2-7}$

$$\int_{0}^{\infty}\frac{4x^2}{(x^4+2x^2+2)^2}dx=\frac{\pi}{15}$$ $$\int_{0}^{\infty}\frac{4x^2}{[(1+(1+x^2)^2]^2}dx=\frac{\pi}{15}$$ $u=\tan(z)$ $\rightarrow$ $du=\sec^2(z)$ $u$ $\rightarrow ...
2
votes
0answers
54 views

Does $\int fdg=0$ for fixed positive $f$ imply $g$ is constant?

The question formulated in the title is not very rigorous, so let me be more specific. Assume that $f:[a,b]\to\mathbb (0,\infty)$ is a given positive, continuous and strictly decreasing function, and ...
1
vote
1answer
40 views

About the relation between $\int_{0}^{\infty}\frac{\cosh((\alpha-\beta)x)}{(\cosh(x))^{\alpha+\beta }} \,dx$ and $B(\alpha,\beta)$

How do I get from $$B(\alpha,\beta)=\int_{0}^{1}{ t^{ \alpha -1} (1-t)^{ \beta -1} \,dt}$$ to: $$2^{2- \alpha - \beta } \int_{0}^{ \infty}{ \frac{ \cosh (( \alpha - \beta )x)}{( \cosh(x))^{ ...
1
vote
2answers
61 views

Integrating the floor of a function

The integral that I'm trying to simplify is this: (both $x$ and $c$ are natural numbers, if that helps) $$F(x,c)=\int_0^c \left(\left\lfloor 10^{-\lfloor t \rfloor} x \right\rfloor -10\left\lfloor ...
5
votes
2answers
90 views

Frullani 's theorem in a complex context.

It is possible to prove that $$\int_{0}^{\infty}\frac{e^{-ix}-e^{-x}}{x}dx=-i\frac{\pi}{2}$$ and in this case the Frullani's theorem does not hold since, if we consider the function $f(x)=e^{-x}$, we ...
1
vote
2answers
80 views

Prove that $\int_{0}^{1}{\ln(x) \over 2-x}dx={\ln^2(2)-\zeta(2)\over 2}$

$$I=\int_{0}^{1}{\ln(x) \over 2-x}dx={\ln^2(2)-\zeta(2)\over 2}$$ $$(2-x)^{-1}={1\over2}\left(1-{x\over 2}\right)^{-1}$$ Using binomial series here $$(2-x)^{-1}={1\over2}\left(1-{x\over ...
-3
votes
0answers
151 views

Integration practice for the beginners

Beginners in calculus may enjoy the following problem: Compute the following integral: $$\int_0^1 \int_0^1 \frac{\displaystyle(1+y) ...
0
votes
0answers
18 views

Closed form for $\int\frac{\left((x + i) \beta\right)^\beta x^{\beta - 2}}{(x^2 + 1)^\beta} \exp\left(-\frac{\alpha}{A x}\right) \, \mathrm{d} x$

The following integral comes up in the solution of a differential equation when solved by Maple: $$ \begin{equation} \int\frac{\left((x + i) \beta\right)^\beta x^{\beta - 2}}{(x^2 + 1)^\beta} ...
0
votes
3answers
44 views

How should I try to evaluate the integral $\int_a^b \sqrt{1 + \frac{x^2}{r^2 - x^2}} \; dx$

I've tried to evaluate $\displaystyle\int_{-r}^r \sqrt{1 + \frac{x^2}{r^2 - x^2}} \; dx$ on my own, but I have encountered a problem I cannot get around. The indefinite integral ...
0
votes
3answers
28 views

Explication on how obtaining $\int \langle \nabla w, \nabla w\rangle = \lambda \int \langle w, w\rangle$

Could anyone is able to explain to me how to obtain $\int \langle \nabla w, \nabla w\rangle = \lambda \int \langle w, w\rangle$ related to user7530's comment in the question : Rayleigh quotient ...
4
votes
1answer
55 views

Can I integrate an asymptotic expression?

Suppose that $y(x; \epsilon)$ is a real-valued function of $x \in [a,b] \subset\mathbb{R}$ depending on a real parameter $\epsilon$, and that \begin{align} \int_a^b dx \ y(x; \epsilon) =& 1 ...
1
vote
2answers
30 views

Parameters leading to an elementary integral

For which values of $a,b$ the following integral is an elementary function, and which elementary function? $$\int \frac{x^2+ax+b}{(x-1)^2}\,e^x\, dx$$ I tried to solve this integral but it ...
0
votes
0answers
9 views

Convolution of matrix coefficients from inequivalent representations

Suppose $\delta_1,\delta_2$ are two inequivalent representations of a compact Lie group. Let $dy$ be the normalised Haar measure and define convolution for functions $f,g:G\rightarrow \mathbb{C}$ ...
-1
votes
1answer
35 views

Tic-Tac-Toe Method (Proof by induction) [on hold]

enter image description hereIs there any proof of Tic-Tac-Toe Method by induction?
0
votes
5answers
46 views

First Year Calculus Problems

I'm doing a few past exam papers for my calculus test. I came across a few problems which I thought would be worth asking about. 1.) Consider the function $f(x) = \begin{cases} x+1, & \text{if ...
6
votes
2answers
105 views

My conjecture $\int_{0}^{1}{x^n-1 \over \ln(x)}dx=\ln(n+1)$ [duplicate]

$$\int_{0}^{1}{x^n-1 \over \ln(x)}dx=\ln(n+1)$$ Let deal with case $n=1$ $$I=\int_{0}^{1}{x-1 \over \ln(x)}dx=\ln(2)$$ $u=\ln(x)$ $\rightarrow du=\frac{1}{x}dx$ $x \rightarrow 1 ,u=0$ $x ...
0
votes
3answers
23 views

How to setup this triple integral

Let $D$ be the region in $\mathbf R^3$ that lies above the surface $z=x^2+y^2-1$and below the surface$z=\sqrt{1-x^2-y^2}$. Sketch the region of $D$ together with its projection on the $XY$-plane ...
0
votes
3answers
165 views

Stuck in integration problem [on hold]

Could someone give me some hint or show me how to calculate this integration? $$\huge{\displaystyle\int_{-\infty}^\infty}z^2e^{-\frac{z^2}{2}}\ dz$$Thanks in advance.
2
votes
2answers
68 views

Can I integrate this without using Cauchy's integral formula

How can I integrate this? $$\frac{1}{2 \pi i} \oint_c \frac{e^z}{z(1-z)^3}dz $$ C is any simple path which encloses 0 and 1. And can it be done without using Cauchy's integral formula?
3
votes
1answer
658 views

Approximating integrals with step functions

For $f \colon [1,2] \to \mathbb{R}$ , $f(x) = 1/x$, Choose a sequence of step functions $\phi_n$ approximating $f$ with partition $P_n := [r/n : n < r < 2n]$ to show that $ 1/(n+1) + \cdots + ...
0
votes
1answer
33 views

Integral $\int_{1}^{\infty}\frac{-4}{(2 \cos{x} - 2) x^3}\ \mathrm dx.$

I'm not sure how to proceed with the following integral: $$I=\int_{1}^{\infty}\frac{-4}{(2 \cos{x} - 2) x^3}\ \mathrm dx.$$ Mathematica could not find a closed form solution for it and I really have ...
27
votes
1answer
3k views

Integrating $\int^{\infty}_0 e^{-x^2}\,dx$ using Feynman's parametrization trick

I stumbled upon this short article on last weekend, it introduces an integral trick that exploits differentiation under the integral sign. On its last page, the author, Mr. Anonymous, left several ...
2
votes
0answers
18 views

The integral is the area under the curve. Is there a similar notion for stochastic integrals?

As discussed in the answers to this question, the integral is defined to be the (net signed) area under the curve. The definition in terms of Riemann sums is precisely designed to accomplish this. ...
7
votes
4answers
190 views

Prove $\int_{0}^{\infty}\frac{2x}{x^8+2x^4+1}dx=\frac{\pi}{4}$

$$\int_{0}^{\infty}\frac{2x}{x^8+2x^4+1}dx=\frac{\pi}{4}$$ $u=x^4$ $\rightarrow$ $du=4x^3dx$ $x \rightarrow \infty$, $u\rightarrow \infty$ $x\rightarrow 0$, $u\rightarrow 0$ ...
2
votes
2answers
67 views

Evaluate this $\int_{0}^{\infty}{x^s-1 \over x-1}\cdot{1-e^{-ax} \over 1-e^{ax}}dx$

$$\int_{0}^{\infty}{x^s-1 \over x-1}\cdot{1-e^{-ax} \over 1-e^{ax}}dx$$ $$\int_{0}^{\infty}{x^s-1 \over x-1}\cdot{1-e^{-ax} \over 1-e^{ax}}\cdot{e^{ax}\over e^{ax}}dx$$ $$\int_{0}^{\infty}-{x^s-1 ...
1
vote
1answer
396 views

proving that the graph of a function is of Jordan measure zero

Let $f$ be an integrable function from $B$ to $[0,\inf]$ where $B$ is a sphere in $\mathbb{R^n}$. Exercise: For $f$ and $B$, the graph $$ \Gamma=\{(x,f(x)):x\in B\} \subset \mathbb{R}^{n+1} $$ is of ...
2
votes
0answers
29 views

Convolution of a function and its inverse

I want to calculate the convolution of a function and its inverse, $$f(t) * f^{-1}(t)$$ e.g. $f(t)=1/(t-2i)$ I've heard that the answer can be a delta function. What requirements are necessary for ...
1
vote
1answer
17 views

$\int(\int\phi(a-z)dz)dz=\Phi(a-z)$

Lets assume $\phi(a-z)$ is integrable. Can I conclude that the following integral $$\int\left(\int\phi(a-z)dz\right)dz$$ Can be expressed by a function $$\Phi(a-z).$$ So in result: ...
1
vote
1answer
51 views

Integral representation of the Beta function

How do I get from $ \int_{0}^{1}{t^{x-1} (1-t)^{y-1}\,\mathrm dt} $ to $ \displaystyle\int_{0}^{\infty}{ \frac{t^{x-1} + t^{y-1}}{(1+t)^{x+y}} \,\mathrm dt} $ ? I have tried different change of ...
3
votes
1answer
97 views

How to integrate $\int{1\over \sqrt{x^2-1}}\mathrm d x$ another technique without use trigonometry

How can I integrate $\int{1\over \sqrt{x^2-1}}\mathrm d x$ without using trygnometry? I mean using other methods like substituition, integration by parts (I tried these two but I think I am not seeing ...
1
vote
2answers
70 views

If $f$ is continuous with $ \int_0^{\infty}f(t)\,dt<\infty$ then which are correct?

Let $f:[0,\infty)\to [0,\infty)$ be a continuous function such that $\displaystyle \int_0^{\infty}f(t)\,dt<\infty$. Which of the following statements are true ? (A) The sequence $\{f(n)\}$ is ...
13
votes
4answers
196 views

How to evaluate $\int_0^1\frac{\ln(1-2t+2t^2)}{t}dt$?

The question starts with: $$\int_0^1\frac{-2t^2+t}{-t^2+t}\ln(1-2t+2t^2)dt\text{ = ?}$$ My attempt is as follows: $$\int_0^1\frac{-2t^2+t}{-t^2+t}\ln(1-2t+2t^2)dt$$ ...
2
votes
2answers
599 views

Numerical approximation of the modified Bessel function $I_0$ with radical argument for integration purposes

I have to numerically calculate the following definite integral $$\int_{\alpha}^{\beta}I_0(a\sqrt{1-x^2})dx$$ for different values of $\alpha$ and $\beta$, where $a$ has a value of, say, $30$. I'm ...
0
votes
0answers
43 views

how to get this integration?

How do I get from $ \int_{0}^{\infty}{\frac{t^{y-1}}{(1+t)^{x+y}}\,\mathrm dt} $ to $ \displaystyle \frac{1}{2} \int_{0}^{\infty}{\frac{t^{x-1} + t^{y-1}}{(1+t)^{x+y}} \,\mathrm dt} $ ?. I have tried ...
-2
votes
2answers
25 views

Double integration of Greatest integer function [on hold]

Integration of double integral of. $$ \int_0^2\int_0^{y-2} \lfloor x + y \rfloor \,dx\,dy $$
0
votes
0answers
62 views

Help with Change of Variable for the function $f(x,y)=e^{\frac{x}{2x+3y}}$

Let $D$ be the open triangle with the vertices $(0,0), (3,0), (0,2)$. For $f(x,y)=e^{ \frac{x}{2x+3y}}$ show that $f$ is integrable on $D$ and prove that $\iint_Df(x,y)dxdy=6\sqrt{e}-6$. I was able ...
12
votes
4answers
687 views

Double obstructing wall problem, what is the optimal walk path and length?

Every day, you walk from point A to point B which are exactly $2$ miles apart straight line distance, however, each day, there is a $50$% chance of there being an obstructing wall perpendicular to the ...