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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3
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0answers
11 views

Probability density on the sphere and change of variables

This question is related to another my question: Solving nonlinear system I have a random vector $X=(X_1,X_2,X_3)$ with a density function $f_X(x)$. Now consider $Y=g(X)=X/\|X\|$. What is a density ...
2
votes
1answer
29 views

Finding a Solution to an Integral Equation

I want to solve the following integral equation: $$ u(t) = \int_t^T a(s) ds + \int_t^T b(s)u(s) ds , $$ with $a, b, u$ being functions from $[t,T] \rightarrow \mathbf{R} $. I transformed the ...
3
votes
2answers
96 views

A problem in integration.

As you know from basic trigonometry that $\sin(2x) = 2\sin(x)\cos(x)$. If you integrate both sides with respect to x, one finds $$\int \sin(2x) \ dx = -\frac{1}{2}\cos(2x)+c$$ on the left hand side ...
1
vote
0answers
18 views

Proof of additivity of domain for definite integrals

I would like to prove the following theorem: Theorem If $c \in (a,b)$ and $f$ is integrable on $[a,c]$ and $[c,b]$, then $f$ is integrable on $[a,b]$ and $$\int_{a}^{b}f = \int_{a}^{c}f + ...
0
votes
2answers
51 views

Analytic integration of this function

Integrate \begin{equation} \int{\frac{1}{(1-\frac{a}{r}-b r^2)}} \, \mathrm{d}r \end{equation} where $a$ and $b$ are constants.
1
vote
1answer
25 views

Integral related to Poisson kernel

$\textbf{Problem}$: Find the value of the integral $$I=\int_{-\infty}^0 P.V.\frac{1}{\pi}\int_{-\infty}^{\infty} \frac{\partial f}{\partial y} \frac{(x-y)}{(x-y)^2+z^2} \ dy \ dz,$$ with $f$ a ...
3
votes
4answers
83 views

Expressing the integral in terms of the original variable

In evaluating the integral: $$ \int{dx\over(a^2-x^2)^{3/2}} $$ or $$ \int{dx\over(a^2-x^2)^{1/2}\ (a^2-x^2)}$$ Let $ x=a\sin\theta $ and $ dx=a\cos\theta\ d\theta $. Then $$ \int{{a\cos\theta\ ...
3
votes
4answers
814 views

Math Subject GRE 1268 Question 55

If $a$ and $b$ are positive numbers, what is the value of $\displaystyle \int_0^\infty \frac{e^{ax}-e^{bx}}{(1+e^{ax})(1+e^{bx})}dx$. A: $0$ B: $1$ C: $a-b$ D: $(a-b)\log 2$ E: ...
1
vote
1answer
51 views

Inequality on integrals of continuous functions: $\int_0^1 f^2(x)\,dx \geq \left(\int_{0}^{1} f(x) \,dx\right)^2$

Let $f\colon [0, 1] \to \mathbb{R}$ be a continuous function. How to prove $$\int_0^1 f^2(x)\,dx \geq \left(\int_{0}^{1} f(x) \,dx\right)^2$$ (I'm not getting anything.. any hint is appreciated)
2
votes
0answers
37 views

Problem with a step involving a type of Riemann integration

I am reading this text, , and I find it unclear how the ratio of the considered rectangle's area to its length tends to become the derivative of a function $S$ as the lenght of the considered ...
2
votes
3answers
46 views

compute temporal average of $\sin(\omega_0t+\Phi)\sin(\omega_0t+\omega_0\tau+\Phi)$

assuming that $\Phi$ is uniformly distributed over $(0,2\pi)$ compute: $$E[\sin(\omega_0t+\Phi)\sin(\omega_0t+\omega_0\tau+\Phi)]$$ I have solved the problem as continues: $$\begin{align} ...
1
vote
0answers
56 views

An integral that I cannot simplify.

Good day, esteemed students of mathematics! I have been trying to prove that the convolution of $2q$ Gaussian probability distributions is another $q$ Gaussian probability distribution with the same ...
1
vote
0answers
75 views

Integrating functions with $x^3$

After learning the integration of various functions with $x^2$ involved, I was given the following integration, as a challenge: $$\sqrt{1+x^3}$$ I tried various methods - too long to even try and ...
2
votes
2answers
62 views

Is $\int_{x=0}^1\int_{y=0}^1\int_{z=0}^1 \frac{1}{(x-y)^2 (y-z)} dx dy dz$ finite?

My question is in the title : How could I prove that $$ \int_{x=0}^1\int_{y=0}^1\int_{z=0}^1 \frac{1}{(x-y)^2 (y-z)} \ \text{d}z \ \text{d}y \ \text{d}x $$ is finite (if it is) ? Thank you by ...
0
votes
2answers
63 views

Compute the integral over the volume of a torus,

In $\mathbb R^3$, let $C$ be the circle in the $xy$-plane with radius $2$ and the origin as the center, i.e., $$C= \Big\{ \big(x,y,z\big) \in \mathbb R^3 \mid x^2+y^2=4, \ z=0\Big\}.$$ Let $\Omega$ ...
1
vote
2answers
32 views

Evaluating the bounds for a triple integral

I've working on the problem: Evaluate $\iiint_Q$ $1/(x^2 + y^2 + z^2)$, where Q is the solid region ABOVE the xy-plane (and we must do this in spherical coordinates). What I've done thus far is ...
0
votes
0answers
36 views

Confused on what the question is asking… Integration and Riemann sums

So I know how to integrate, and I understand Riemann sums via Right hand, left, etc but this question I am trying to answer has me baffled. I think its because its trying to be generic in the f(mx) ...
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votes
1answer
49 views

Is there an equation for the integral? [on hold]

For example, the equation of a derivative would just be f(x+h)-f(x)/h What would the equation of an integral be? Or does it just go backwards.
1
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0answers
22 views

Will this method always find the maximum of a positive-definite function?

Definitions: A real-valued, continuously differentiable function $f$ is positive definite on a neighborhood of the origin, $D$, if $f(0) = 0$ and $f(x) > 0$ for every non-zero $x \in D$. ...
-1
votes
0answers
21 views

How do you calculate an area enclosed by four tangents by using the integration method?

For example, make it $y=3x-6$, $y=3x-15.48$, $y=-0.25x+1.25$, and $y=-0.25x-1.06$. It's been taken by finding the tangent line of a curve $y=(x-2)(x-3)(x-5)$.
0
votes
1answer
20 views

To what fractional Sobolev spaces does the step function belong? (Sobolev-Slobodeckij norm of step function)

I'm new to fractional Sobolev spaces and I'm curious about the regularity of some simple functions like e.$\,$g. step functions in order to understand these spaces better. In more detail, for $\Omega ...
4
votes
0answers
55 views

Integral $\int z^2\Re(J_1(z))dz$

$$ \int x^2 \, \Re\left[{J_1(a x)}\right]dx=\frac{1}{a^3}\int z^2 \Re\left[\frac{z}{2}\sum_{k\geq 0} \frac{(-1)^k}{k!\Gamma(k+2)} \left(\frac{z}{2}\right)^{2k}\right]dz $$ where $a\in \mathbb{C}$ and ...
-2
votes
1answer
59 views

Integrations Question Answer [on hold]

I need your help to evaluate this integral: $$\int\frac {x^5}{1+x^6}~\mathrm{d}x $$ Any help will be appreciated!
11
votes
1answer
151 views

Could it possibly have a nice closed form? $\int _0^1\int _0^1\frac{x y}{(x+1) (y+1) \log (x y)}\ dx \ dy$

Using multiple integrals it's not hard to show that the present integral reduces to some integral over squared digamma functions, but then things become harder. How would you tackle the problem? ...
1
vote
1answer
37 views

integrating product of PDF and CDF

I am trying to show that the following integral: $$ \int_{-\infty}^a F(x)~f(x)~dx = \frac{F(a)}{2!} $$ Where $F$ is the cumulative distribution function of some continuous random variable X, and $f$ ...
5
votes
2answers
91 views

Problem 7 IMC 2015 - Integral and Limit

I'm trying to solve problem 7 from the IMC 2015, Blagoevgrad, Bulgaria (Day 2, July 30). Here is the problem Compute $$\large\lim_{A\to\infty}\frac{1}{A}\int_1^A A^\frac{1}{x}\,\mathrm dx$$ ...
-5
votes
2answers
37 views

solve this linear equation

Using linear differential equation, solve the following equation $( y \log (x)-2) y \textrm{d} x =x \textrm{d}y$. Source: "higher engineering mathematics by grewal"
3
votes
5answers
84 views

Elegant solution for $\int {\frac{\cos(y)}{\sin^2(y)+\sin(y)-6}}dy$

I have the following integral: $\int {\frac{\cos(y)}{\sin^2(y)+\sin(y)-6}}dy$ I already know the solution, but it needs three substitutions. Is there a simpler, more elegant way to go about this?
1
vote
1answer
40 views

Take an example of Integrate of root

I want to solve an example like this : $\int_{0}^{4}\sqrt{4^2-x^2}\ dx$ according to this equation :$$\int \sqrt{a^2-x^2}\ dx= ...
1
vote
1answer
18 views

Finding potential function of $\vec F =xy^2 \hat i +y x^2 \hat j$

$$\vec F =xy^2 \hat i +y x^2 \hat j$$ My attempt: $$P=U_{x}=xy^2$$ $$Q=U_{y}=x^2y$$ $$\Longrightarrow U=\int P dx=\frac{x^2}{2}y+C(y)$$ $$ U_{y}=\frac{x^2}{2}+C'(y)=Q=x^2y$$ ...
1
vote
1answer
13 views

Anti-deriving composition of a non-linear activation function on Fourier series?

My pea-brain is not commensurate with the big words in the title. But I'm working on a project where I need to compute definite integrals of the composition $f(g(x))$, where $f(x)$ is any non-linear ...
8
votes
1answer
99 views

Integral $\int_0^\infty\text{Li}_2\left(e^{-\pi x}\right)\arctan x\,dx$

Please help me to evaluate this integral in a closed form: $$I=\int_0^\infty\text{Li}_2\left(e^{-\pi x}\right)\arctan x\,dx$$ Using integration by parts I found that it could be expressed through ...
0
votes
0answers
36 views

Evaluating triple integrals that are bounded

I'm slowly learning how to bound triple integration problems, but this one has me a little confused. $\iiint_D(x+2y)dV$, where D is bounded by the parabolic cylinder, $y = x^2$, and the planes x=z, ...
4
votes
1answer
69 views

Integral written as the integral of a measure

Let $(X,\mathcal M,\mu)$ be a measure space and let $f\in L^1(X,\mu)$ be a positive function. Show that $$\int_X f \, d\mu=\int_{(0,\infty)} \mu(\{f>t\}) \, dt.$$
1
vote
1answer
34 views

Finding the volume of a solid region

I'm trying to find the volume of the solid region inside the sphere $x^2+y^2+z^2=4$, and the upper nappe of the cone $z^2=3x^2+3y^2$ (I only have to set up the triple integral itself, not evaluate ...
2
votes
1answer
54 views

Difference between line integrals in complex analysis and real analysis,

The formula in complex analysis is $$\int f(\gamma(t))\cdot(\gamma'(t)dt$$ and the formula in the real variable setting, for a gradient field, is: $$\int F\cdot dr$$ $$=\int f_x\,dx + f_y\,dy + ...
1
vote
1answer
21 views

Finding the work from $(0,0)\to(1,1)$ of $\vec F(x,y)=xy^2\hat i+yx^2\hat j$

I need to find the work from $(0,0)\to(1,0)\to(1,1)$ of the following vector field:$\vec F(x,y)=xy^2\hat i+yx^2\hat j$ My attempt: $$\oint_{c}\vec F d\vec r=\int_{(0,0)\to (1,0)}\bigg(xy^2\; dx ...
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votes
1answer
37 views

How to find triple integral of the following question?

I've been trying to solve that question and over and over again, I get answer of: Where as the online integral solver gives an answer of: I am really confused that If I am correct or the online ...
7
votes
0answers
107 views

Wanted: Simple integration theory

Supposing we want to formulate a very primitive theory of integration, the only requirement being that all continuous functions $[a, b]\longrightarrow\mathbb{R}$ be integrable. What is the simplest ...
2
votes
2answers
47 views

Evaluate $\oint_{C} e^{-x} \sin y \;dx+e^{-x} \cos y\;dy$

I need to evaluate the following integral using Green's theorem $$\oint_{C} e^{-x} \sin y \;dx+e^{-x} \cos y\;dy$$ $C$: from point $E \to F\to G\to H$ ...
5
votes
1answer
33 views

Gauss Hermite Integration of 1/(1+x^2)

I'm trying to learn Gauss Hermite Integration and was manually try to calculate the value of integral of $\frac{1}{1+x^2}$ from $-\infty$ to $+\infty$ The exact answer is simply $\pi$ ($\approx$ ...
1
vote
4answers
51 views

Trigonometric substitution and triangles

I'm learning trigonometric substitutions and am having a bit of trouble understanding the intuition behind the conversions (why do most use secant?). If you could explain the conversions geometrically ...
1
vote
2answers
73 views

Convergence, Integrals, and Limits question

Let $f: [0,\infty)\to \Bbb R$ be a positive,decreasing monotonic function. Prove the following statement for every a>0 providing the integral on the right side converges. First I managed to ...
1
vote
1answer
26 views

Distribution Technique Question of two independent Exponential Distributions

If $X_1$ and $X_2$ are two independent random variables having exponential densities then $f(x_1,x_2)$ is defined as $$f(x_1,x_2)=\exp(-(x_1+x_2))\,{\bf 1}_{(0,\infty)}(x_1){\bf ...
3
votes
3answers
66 views

Calculate derivative of integral

I tried to calculate the derivative of this integral: $$\int_{2}^{3+\sqrt{r}} (3 + \sqrt{r}-c) \frac{1}{2}\,{\rm d}c $$ First I took the anti-derivative of the integral: ...
2
votes
2answers
74 views

How to find integral of sqrt(sinx cosx)

I have been working on days to find the integral of the following question: $$ \int\sqrt{\sin x\cos x}\,dx $$ Any anyone please help in finding the solution of that question?
1
vote
1answer
64 views

Can “Integration by parts” be used to integrate any function?

I am having hard time understanding integration by substitution method so can I relay on integration by parts?
1
vote
1answer
20 views

What is the maximum value of work done by this force field?

An object moves in the force field $F=yz\hat{i}+zx\hat{j}+xy\hat{k}$ starting at the origin and ending at some point $A(\xi,\eta,\zeta)$ that lies on the surface ...
-1
votes
1answer
37 views

Center of gravity of a hollow or solid semi sphere [on hold]

Find the center of gravity of a hollow semi sphere with radius"a" through integration.Through that(using the above answer) deduce the center of gravity of a semi solid sphere(with radius a) is "3a/8" ...
0
votes
0answers
25 views

Is this upper bound ok to use when bounding the error between the Riemann sum and its integral?

I found this on some class notes, which gives several different estimates of the error term, when going from the Riemann sum to its corresponding Riemann integral: $$\frac{b-a}{n}[f(b)-f(a)]$$ Does ...