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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6 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 ...
2
votes
1answer
46 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 ...
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0answers
30 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, ...
3
votes
1answer
58 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 $$\int_X f \, d\mu=\int_{(0,\infty)} \mu(\{f>t\}) \, dt.$$
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1answer
32 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
51 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 + ...
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1answer
20 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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1answer
34 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 ...
6
votes
0answers
76 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
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2answers
44 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
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1answer
28 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 $\pi$ ($\approx$ ...
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4answers
50 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
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2answers
67 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
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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
57 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
70 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?
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1answer
62 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?
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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 ...
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1answer
26 views

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

Find the center of gravity of a hollow semi sphere trough integration and through that prove the center of gravity of a semi solid sphere(with radius a) is $\frac{3a}{8}$
0
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0answers
23 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 ...
4
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3answers
90 views

Show that this difference goes to zero,

$$\frac{1+\sqrt{2} + ... + \sqrt{N}}{N} - \frac{2}{3}\sqrt{N} \to 0.$$ The hint given in the question is this: choose appropriate Riemann sums and estimate the approximation error. My current work: ...
2
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0answers
48 views

Green's theorem application

Problem Determine all circles $\mathcal C$ on $\mathbb R^2$ such that $$\int_{\mathcal C}-y^2dx+3xdy=6\pi$$ My attempt at a solution If I call $P(x,y)=-y^2$ and $Q(x,y)=3x$, then I can apply ...
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2answers
45 views

Integrate area of function over a tetrahedron

I actually attempted to enlist my professor help on this problem, but what he said was quick and I must not have written everything down because I cannot understand how this problem is supposed to be ...
0
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1answer
75 views

Possible values of a $f(x)=(ab-b^2-2)x+\int_{0}^{x} x^2(\cos^{4}t+\sin^{4}t)\mathrm{d}t$

Suppose $f(x)=(ab-b^2-2)x+\int_{0}^{x} x^2(\cos^{4}t+\sin^{4}t)\, \mathrm{d}t$ is a decreasing function of $x$, $x$ is a real number. What are the possible values of $a$? $b$ is independent of $x$. I ...
0
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1answer
45 views

Evaluations of a Definite Integral with cosine function

How do you evaluate this integral? Does it involve an elliptical integral? What technique do I use to evaluate this integral? $$\int _{ 0 }^{ 2\pi }{ \sqrt { 5-4\cos { \theta } } d\theta } $$
3
votes
1answer
28 views

Convergence of Integrals of Exponential Functions

Let $f$ be a non-negative real valued function on $[a,b]$, and let $p:[a,b]\to(1,\infty)$ such that $f^p\in L^1([a,b])$. Let $p_n:[a,b]\to(1,\infty)$ be a (uniformly bounded) sequence of ...
0
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0answers
20 views

Rankine Hugoniot, taking limits

I have seen two different derivations of the Rankine Hugoniot jump conditions across a shock s(t) in the xt-plane. I present a summary of the two different derivations and then post my question in ...
0
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0answers
25 views

Yet another asymptotic series that needs to be analyticaly extended

Let $A>0$ and $1\le \mu \le 2$. Consider a following definite integral: \begin{equation} {\mathcal I}(A,\mu) := Re\left[\int\limits_0^\infty e^{-(k A)^\mu}\frac{\left(\gamma+\Gamma(0,\imath ...
0
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1answer
26 views

How to differentiate with respect to component of a vector?

Let $\vec{\alpha}=\frac{m(\vec{x})}{x^2}\vec{x}$ where $\vec{x}=(x_1,\,x_2)$. In a book I read in Eq.(3.24), it was given that $$ \frac{\partial \alpha_1}{\partial x_1}=\frac{d m}{d ...
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0answers
27 views

To determine the points of $\Bbb R^2$ at which $(i) f_x$ exists, $(ii) f_y$ exists.

Let $f : \Bbb R^2 → \Bbb R$ be defined by $f(x, y) := x^2 + y^2$ if $x$ and $y$ are both rational, and $f(x, y) := 0$ otherwise. To determine the points of $\Bbb R^2$ at which $(i) f_x$ exists, $(ii) ...
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1answer
31 views

Doubling measure of an annulus

Recall that a doubling measure is a measure with the additional requirement that: $$\mu(B_{2R})\le C_\mu \mu(B_R)$$ for some contstant $C_\mu$. While solving some esercises related to doubling ...
6
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0answers
45 views

How to solve this definite Integral containing $E_{1}${.}!

The integral is: $$\int_{N}^{\infty}\frac{E_{1}(cz+d)}{az+b}e^{-pz}dz$$ where, $E_{1}${.} is the exponential integral, and $$a>0,\ b>0,\ c>0,\ d>0,\ p>0,\ N>0.$$ This is similar ...
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2answers
49 views

How to calculate an elementary integral

How do you calculate $$\int\dfrac{2 du}{(u^2+1)^2}$$ It does not seem too difficult but I do not know which method to use.
7
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1answer
119 views

Closed-form of $\int_0^1\left(\frac{\left(x^2+1\right)\arcsin(x)}{\sqrt{1-x^2}}+2x\ln\left(x^2+1\right)\right)\frac{\ln x}{x^3+x}\,dx$

I've conjectured the following closed-form: $$ I = \int_0^1\left(\frac{\left(x^2+1\right)\arcsin(x)}{\sqrt{1-x^2}}+2x\ln\left(x^2+1\right)\right)\frac{\ln x}{x^3+x}\,dx = -2\,G\,\ln2, $$ where $G$ is ...
1
vote
1answer
23 views

Find the region of integration as defined by two paraboloids

I've been given the following problem, and I'm completely unsure how to go about solving it. $$ \text{Find the volume of the solid enclosed by the}\\ \text{paraboloids } z = 16 \left( x^{2} + y^{2} ...
0
votes
1answer
26 views

How to evaluate dh/dt giving dV/dt?

Water evaporates from an open bowl of unspecified shape at a rate proportional to the area ofthe water surface; that is, $$\frac{dV}{dt} = -cA(h)$$ where V is the volume of water, A(h) is the area of ...
0
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2answers
66 views

Help with Calculate integral

Find $\int^a_0 \dfrac{3x^2-ax}{(x-2a)(x^2+a^2)} dx$ I tried using partial fractions and the substitution $u=a-x$ but I haven't made any real progress. Please help.
4
votes
1answer
64 views

Calculation of integral using two different methods? [on hold]

Find $$\int \dfrac{x^3}{(x^2+1)^3}dx$$ in two different ways, first using the substitution $u=x^2+1$ and then using the substitution $x=\tan \theta$. I managed to do both of these but the answer is ...
0
votes
2answers
25 views

Finding the bounds for a triple integration

I'm currently working on a problem stating: $\iiint_Q y*dV$, where Q is the solid that lies between the cylinders $x^2+y^2=1$ and $x^2+y^2=4$, above the xy-plane, and below the plane z=x+2. My ...
3
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3answers
190 views

Indefinite integration: $\int x^{x^2+1}(2\ln x+1)dx$

Find the value of the integral: $$\int x^{x^2+1}(2\ln x+1)dx.$$ My attempt: I tried by using integration by parts, but not working since $x^{x^2+1}$ keeps coming again and again. Then I tried putting ...
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votes
1answer
17 views

Find area of exponential function over box-like region

The problem doesn't seem like it should be too difficult, I have a box-like region $B$ defined as: $$ \begin{align} 0 \le x \le 1\\ 0 \le y \le 3\\ 0 \le z \le 2 \end{align} $$ And the function to ...
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votes
2answers
36 views

How to solve this probability formulation? [on hold]

$\int_{200}^{250} P(a=x \land 450-x \leq b \leq 250)\space dx$, where $a$ and $b$ are uniformly distributed random variables on $(0,250]$ and $(10, 250]$ respectively.
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2answers
42 views

Integration by Partial Fractions $\int\frac{1}{(x+1)^3(x+2)}dx$

I'm trying to do a problem regarding partial fractions and I'm not sure if I have gone about this right as my answer here doesn't compare to the answer provided by wolfram alpha. Is it that I can't ...
2
votes
1answer
52 views

Finding the area under the cycloid $x=t-\sin (t),\;y=1-cos (t)$

I need to find the area under the cycloid $x=t-\sin (t),\;y=1-cos (t)$ above axis and between $x=0,x=2\pi$ using $\underline{\text{Green's theorem}}$ I found in Wikipedia this evaluation: ...
1
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1answer
47 views

Can someone help me understand this: integrating over a discrete set of points yields 0 under Lebesgue integral?

Suppose I had some linear function $f(x)$ and then I sampled the function over the integers to form $f(n)$, what would be the evaluation of the Lebesgue integral of $\int_\mathbb{Z_+} f(n) d\mu$? For ...
2
votes
2answers
47 views

Finding the volume of a solid bounded by a sphere and a paraboloid

I am working on a problem that requires me to find the volume of the solid bounded by the sphere $x^2 + y^2 + z^2 = 2$ and the paraboloid $x^2 + y^2 = z$. I know that to do this, I must use triple ...
6
votes
3answers
149 views

Closed-form of $\int_0^\infty \frac{1}{\left(a+\cosh x\right)^{1/n}} \, dx$ for $a=0,1$

While I was working on this question by @Vladimir Reshetnikov, I've conjectured the following closed-forms. $$ I_0(n)=\int_0^\infty \frac{1}{\left(\cosh x\right)^{1/n}} \, dx \stackrel{?}{=} ...
0
votes
1answer
62 views

Evaluate $\oint_{C}xy^2dx+2x^2 dy$

$$\oint_{C}xy^2dx+2x^2y dy$$ triangle:$$C=\{(0,0),(2,2),(2,4)\}$$ My attempt: Using Green's theorem $$\oint_{C}\underbrace{xy^2}_{P}dx+\underbrace{2x^2y}_{Q} dy=\iint\bigg(\frac{\partial ...
6
votes
1answer
41 views

Limit behavior of a definite integral that depends on a parameter.

Let $A>0$ and $1\le \mu \le 2$. Consider a following integral. \begin{equation} {\mathcal I}(A,\mu) := \int\limits_0^\infty e^{-(k A)^\mu} \cdot \frac{\cos(k)-1}{k} dk \end{equation} By ...
1
vote
2answers
32 views

How to find the surface area of revolution of an ellipsoid from ellipse rotating about y-axis

Suppose the ellipse has equation $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$. I understand the way to obtain the surface area of the ellipsoid is to rotate the curve around y-axis and use surface of ...