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
votes
1answer
64 views

Integral of $p(x)\operatorname{csch}(x)$

I'd like to calculate the following integral $$\int_{-\infty}^{+\infty}\frac{x^4 \left(\frac 1 {a^2+x^2} +\frac 1 {b^2+x^2}\right)}{\sinh^2(x\pi /c)} \, dx$$ where $a$, $b$ and $c$ are positive ...
0
votes
2answers
79 views

Evaluate $-\int_{0}^{\infty}x\sin(x)\ln(1-e^{-x})dx=2\sum_{n=1}^{\infty}\frac{1}{(n^2+1)^2}$

$$-\int_{0}^{\infty}x\sin(x)\ln(1-e^{-x})dx=2\sum_{n=1}^{\infty}\frac{1}{(n^2+1)^2}$$ Is there a closed form for this integral? I was only able to find the equivalent sum to it(How? using wolfram ...
3
votes
4answers
152 views

Definite integral which does not evaluate

How can I solve the following integral? $$\int_{0}^{2\pi }\frac{\cos\phi \cdot \sin(\theta -\phi )}{1-\cos(\theta -\phi )}\text d\phi $$ I have evaluated this integral on Maple without the limits. ...
0
votes
1answer
13 views

parameterise line segment or function

Find the work done by the force field $F =\langle x^2 , ye^x \rangle $ to move a particle along the curve $x = y^2 + 1$, from $A = (1, 0)$ to $B = (2, 1)$. Can I parameterise this curve as a line ...
1
vote
0answers
44 views

Inverse function theorem and integrals.

Consider this integral: $$I = \int g\left(f(x)\right)dx.$$ Assuming all regularity conditions, by inverse function theorem, $$\frac{df(x)}{dx}=\frac{1}{\left[f^{-1}\right]'\left(f(x)\right)}$$ and ...
-2
votes
0answers
78 views

integral $\int_0^\infty \sin x^2 \, \mathrm{d}x $ [duplicate]

I need help with computing this integral. $$\int_0^\infty \sin x^2 \, \mathrm{d}x $$
-1
votes
1answer
50 views

How to compute the integral $I_{\alpha} $?

Someone has an idea to calculate the following integral $$I_{\alpha} = \int_{0}^{+\infty} t^{-\alpha} (1-a)^{t} dt; \quad 0<a,\alpha<1.$$ Thank you in advance
1
vote
1answer
26 views

How to compute $E[Xr / (Xr +1 - X)] $ where $X$ follows a Beta distibution?

I would like to compute $E[Xr / (Xr +1 - X)] $ where $X$ follows a Beta distribution $Beta(\alpha, \beta)$ with $\alpha, \beta > 1$, $\alpha < \beta$ and $r \in (0,1)$. This is the same as ...
0
votes
3answers
61 views

limit with Riemann integral

Assume $f\colon\mathbb{R}\to\mathbb{R}$ is Riemann integrable on $[0,A]$ for every $A>0$ and $\lim_{x\to \infty}f(x)=1$. Prove that $$ \lim_{t\to 0^+}\ t\!\int_{0}^{\infty} e^{-tx}f(x) dx\ =\ 1 $$
0
votes
1answer
18 views

Parametrizing a line integral vector field

evaluate the line integral where F = < y^2, x, z^2> and C is the curve of intersection of the plane x+y+z=1 and the cylinder x^2+y^2=1. orientated clockwise when viewed from above. I'm a bit ...
0
votes
1answer
28 views

$\overline{\int}_{E\cup F}f=\overline{\int}_{E}f+\overline{\int}_{F}f$ using ${\overline{\int}_{\Bbb{R}^n}}f=\inf\{\int h|\text{integrable }h\ge f\}$

Given $E$ and $F$ are disjoint, I have to show that $\overline{\int}_{E\cup F}f=\overline{\int}_{E}f+\overline{\int}_{F}f$ using ${\overline{\int}_{\Bbb{R}^n}}{f}=\inf\{\int h|\text{integrable }h\ge ...
1
vote
0answers
31 views

Convergence of a double sum to an integral

Suppose $f$ and $g$ are periodic continuous functions on $[0,1]$, what is the limit of following sum for $q\to \infty$ ?: $$ \lim_{q \to \infty}\frac{1}{q^2} \sum\limits_{k =0}^{q-1} \sum\limits_{a ...
0
votes
1answer
22 views

Green's Theorem

Use Green's Theorem to evalutate $$ \int <2y^2 + sqrt(1+x^5) , (5x-e^y)> dr $$ $$ C: x^2+y^2=4 $$ C is positively orientated $$ \int \int (dN/dx - dM/dy) dA $$ $$ = (5 - 4y) dA $$ $$ ...
2
votes
2answers
42 views

Find $\displaystyle\int_0^\pi f_{(x)}dx$ when $f_{(x)}=\displaystyle\int_0^x\frac{\sin t}{t}dt$ using $f_{(\pi)}=\beta$

There's a hint included in this question to use integration by parts. Find $\displaystyle\int_0^\pi f_{(x)}dx$ when $f_{(x)}=\displaystyle\int_0^x\frac{\sin t}{t}dt$ using $f_{(\pi)}=\beta$ Here are ...
3
votes
1answer
27 views

Riemann Integrability v.s General Integrability

In my textbook, the definition for Riemann Integrability is as follows: DEFINITION. A function $f$ is said to be Riemann integrable or more simply integrable on a finite closed interval $[a,b]$ ...
1
vote
1answer
21 views

Converting the integral for the area of a circle from an expression involving $dp$ to one involving $dr$

From the Better Explained website, the integral for calculating the area of a circle with respect to its perimeter is given as $$\int_0^{2\pi r} \frac 12 r dp$$ where $r$ is the radius and $p$ is the ...
0
votes
0answers
17 views

Line integral over a vector field with 2 coordinates?

Find the work done by the force field $F = xi + yj + (xz − y)k$ to move a particle along the curve with parametric equations $r(t) =< t^2 , 2t, 4t^3 >, 0 ≤ t ≤ 1,$ from $A(0, 0, 0)$ to $B(1, 2, ...
-2
votes
1answer
36 views

how can i integrate this integrand?(solid angle of circular loop) [closed]

hi im solving some electrodynamics problem but im troubled with integration. i don`t know how to integrate this integrand analytically. what i want to integrate is ...
0
votes
1answer
36 views

Laplace transform of $\frac{e^{-4t}}{\sqrt{t}}$ [closed]

How one may find the Laplace transform of the following function? $$\frac{e^{-4t}}{\sqrt{t}}$$
0
votes
2answers
32 views

Finding the surface area by rotation of cos(x/2)

I have been asked to find the surface area formed when $y=\cos(x/2)$ is rotated around the $x-$axis from $x=0$ to $\pi$. I understand how to set up the integral, but I am really struggling solving it. ...
0
votes
3answers
44 views

Integration by parts, twice

Use the substition $u=4x-3$ to find $\int\frac{4x}{4x-3}dx$, giving your answer in terms of $x$ $$u=4x-3$$ $$4x=u+3$$ $$\frac{du}{dx}=4$$ $$dx=\frac{1}{4}du$$ $$\int\frac{4x}{4x-3}dx$$ ...
9
votes
5answers
482 views

538.com's Puzzle of the Overflowing Martini Glass - How to compute the minor and major axis of an elliptical cross-section of a cone

FiveThirtyEight.com Riddler Puzzle / May 13 The puzzle goes like this; "It’s Friday. You’ve kicked your feet up and have drunk enough of your martini that, when the conical glass (🍸) is upright, the ...
0
votes
1answer
44 views

How to solve this exp Integral

I am trying to solve the following integral, $$ I = \int_0^\infty \mathrm{e}^{z/2 - {\left(z - \ln a\right)^2}/4b} - \mathrm{e}^{z/2 - {\left(z + \ln a\right)^2}/4b}dz $$ where $a$ and $b$ are some ...
3
votes
3answers
85 views

Differentiation under the integral sign for $\int_{0}^{1}\frac{\arctan x}{x\sqrt{1-x^2}}\,dx$

Hello I have a problem that is: $$\int_0^1\frac{\arctan(x)}{x\sqrt{1-x^2}}dx$$ I try use the following integral $$ \int_0^1\frac{dy}{1+x^2y^2}= \frac{\arctan(x)}{x}$$ My question: if I can do ...
2
votes
1answer
147 views

What does $∫ρ(x)dx=∫ρ(x)ϕ(x)dx$ imply about $ϕ(x)$?

What does $$\intop_{D}\rho(x)dx=\intop_{D}\rho(x)\phi(x)dx$$ imply about $\phi(x)$ for some positive definite function $\rho(x)$ which satisfies $$\intop_{D}\rho(x)dx=M$$ For some nonzero M. ...
0
votes
1answer
50 views

Explain inequality of integrals by taylor expansion

I try to understand why the following inequality holds. $$\left|\int_{|y|<1} e^{iuy}−1−iuy\ \, dy \right| \le \frac{1}{2} \cdot \int_{|y|<1} |uy|^2\ \, dy$$ Due to a hint I'm pretty sure, ...
-1
votes
1answer
35 views

Taylor expansion of function [closed]

I try to figure out how the taylor expansion of the following function looks like, but so far I wasn't successfull: $z↦e^{iuz}−1−iuz$ for $|z|<1$. Who has an idea?
-1
votes
0answers
36 views

Are there more of these symmetrical equalities $\int_{0}^{1}\left(\frac{1+\sqrt{5x}}{2}\right)^3dx=\frac{9}{16}+\phi$ [closed]

$\phi=\frac{1+\sqrt5}{2}$ $\Phi=\frac{1-\sqrt5}{2}$ $$\int_{0}^{1}\left(\frac{1+\sqrt{5x}}{2}\right)^3dx=\frac{9}{16}+\phi$$ $$\int_{0}^{1}\left(\frac{1-\sqrt{5x}}{2}\right)^3dx=\frac{9}{16}+\Phi$$ ...
3
votes
1answer
58 views

Evaluate $\int{\sqrt{a^2 - x^2}}dx$

I'm trying to solve the following integral, but seems these 2 methods led to different answers. I think one of the methods must be incorrect. But why doesn't one of them work? Evaluate ...
1
vote
3answers
36 views

Approximation of an indefinite integral

Consider this integral $$\frac{1}{2d}\int_{-d}^{d}f(x-t) \, \mathrm{d}t$$ When $d$ goes to zero, $$\lim _{d\to 0} \frac{1}{2d}\int_{-d}^{d}f(x-t) \, \mathrm{d}t = f(x)$$ but what is the second ...
2
votes
1answer
28 views

Calculus II: Find the Volume (Shell-Method)

Find the volume of the shape created when rotating the region(s) bounded by $y=\sqrt{x+1}, y=0, x=0, x=1$, about the x-axis. I know this is a rudimentary question. My issue is that I tried to test ...
0
votes
2answers
64 views

Integration of $f(x)^n$, given $f(x)$ is continuous and $0 \le f(x) \le 1$

Given $f(x)$ is continuous and $0 \le f(x) \le 1$, and $\int f(x) dx = g(x)$, is there indefinite or definite integration formula for $$\int f(x)^n dx$$ or an approximation or expansion?
2
votes
2answers
67 views

Integrate $ \int\frac{1+x^2}{(1-x^2)(\sqrt{1+x^4})}dx $

$$ \int\frac{1+x^2}{(1-x^2)(\sqrt{1+x^4})}dx $$ I thought of substituting $ x-\frac{1}{x} $ as $t$ but it gets stuck midway. I am close but I think I need to sustitute something else here.
3
votes
3answers
82 views

Prove that $\int_0^{\infty} \frac{\log (1+x)}{x^2}dx$ is divergent.

Could you please tell me how to prove that $$\int_0^{\infty} \frac{\log (1+x)}{x^2}dx$$ is divergent? I calculated an indefinite integral but I don't know how to prove that it diverges.
0
votes
1answer
30 views

Find the area bounded by a curve by changing variables

Calculate the area bounded by the following formula: $$\left(\frac{x^2}{a^2}+\frac{y^2}{b^2} \right)^2 = \frac{xy}{c^2}$$ where $a,b,c>0.$ I have used changing variable of $x=au$ and $y=vb$ to ...
1
vote
3answers
45 views

Simplifying a definite integral expression

If $a_1,a_2,a_3$ are the three values of a which satisfy the equation $$\int_{0}^ {\pi/2}(\sin x+a\cos x)^3dx-\frac{4a}{\pi-2}\int_{0}^{\pi/2}x\cos x dx=2$$ Find the value of $a_1+a_2+a_3$. Now ...
1
vote
0answers
17 views

Distribution of Normal Mixture of Uniforms

Let's say I know that \begin{align*} X \mid \mu,\sigma & \sim \mathcal{N}(\mu, \sigma^2), \\ \mu & \sim Unif(a,b), \\ \sigma & \sim Unif(c,d). \end{align*} I'd like to know the marginal ...
1
vote
0answers
25 views

What is a Well behaved Integral?

I have been reading a book on neural science but i am stuck in the part of well behaved integral. Exactly what is a well behaved integral and what is its relation with summation? (I am a beginner)
0
votes
0answers
21 views

Line Integral of $y$ and $z$

$$\int_C yz^2\,ds$$ $x = 4t,\,y = 3\sin t,\,z = 3\cos t,\\0\leq t\leq\frac\pi2$ I know how to take integrals like this, but the function is of $y$ and $z$. I usually see them in terms of $x$ and ...
1
vote
0answers
16 views

How is justified the derivation under the integral sign $\frac{d}{d\sigma} \left( \Re\frac{1}{\zeta(s)} \right) $?

Taking $\sigma=\Re s>1$ (this is we take $s=\sigma+it$, $\sigma$ and $t$ real numbers) then the using theknown integral representation for $\frac{1}{\zeta(s)}$, where $\zeta(s)$ is the Riemann ...
-2
votes
1answer
38 views

Integrate $ \int\frac{x-1}{x^2\sqrt{2x^2-2x+1}}dx $ [closed]

$$ \int\frac{x-1}{x^2\sqrt{2x^2-2x+1}}dx $$ I am unable to begin this question. Please give some hints or solve it.
6
votes
1answer
68 views

Find $\lim_{n \to +\infty} \int_{0}^{\infty} e^{-x} (nx - [nx]) dx $ [closed]

Find the limit $$\lim_{n \to +\infty} \int_{0}^{\infty} e^{-x} (nx - [nx]) dx $$ where $n$ is a natural number and $[nx]$ denotes the largest integer that is not greater than $nx$.
0
votes
0answers
11 views

How do you find volume integral of Vector functions? [closed]

Got this question in my exams and have argument over how to solve it? 2xz i-x j+y^2 k over x=0 to 2 y=0 to 6 z=0 to 4
1
vote
2answers
56 views

evaluate the $\int_{0}^1 x\,dx$ using the definition

From the definition -"$f$" is integrable on [a,b] if there exists a number $A$ so that for any $\epsilon > 0$ there exists a $\delta > 0$ such that if the sequence ${X_i}$ from $i=0,n$ is a ...
3
votes
0answers
46 views

Does integration wrt to a differential form always come from a measure?

More precisely, is there an $n$-manifold $M$ with an $n$-form $\omega$ such that there is no measure $\nu$ on $M$ satisfying $$\int f \omega = \int f d\mu $$ for all compactly supported smooth ...
3
votes
1answer
93 views

Integrate $ \int \frac{x^2 + x}{(e^x + x +1)^2}dx $

$$ \int \frac{x^2 + x}{(e^x + x +1)^2}dx $$ I cant think of any substitution to start this question.
1
vote
2answers
36 views

Proving integration formula

I want to prove the integration formula $$ \int \frac {\sqrt {a+bu}}{u} \ du = 2 \sqrt {a+bu}+a \int { \frac {du}{u \sqrt {a+bu}} }. $$ I tried trigonometric substitution (as $u= \frac {a \tan^2 ...
1
vote
5answers
109 views

Integrate $ \int \frac{1}{1 + x^3}dx $

$$ \int \frac{1}{1 + x^3}dx $$ Attempt: I added and subtracted $x^3$ in the numerator but after a little solving I can't get through.
3
votes
3answers
59 views

Integrate $ \int \frac{1+xcos(x)}{x(1-x^2(e^{2sin(x)}))}dx $

$$ \int \frac{1+x\cos x}{x(1-x^2e^{2\sin x})}dx $$ Attempt: I substituted $(1-xe^{2sin(x)})$ by $u$ and tried from there by differentiating it. But I get stuck midway.
4
votes
0answers
44 views

Complex Contour Integral Involving Arg(z)

My question is regarding the following complex integral: $$\int_\gamma\frac{\operatorname{Arg}(z)}{z} dz$$ where $\gamma$ is the curve defined by:$\quad$ $\gamma(t) = e^{it}, 0\leq t\leq ...