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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4
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0answers
61 views

How to get to the closed form of $\int_{-\infty}^{\infty} \frac{x^2e^x}{(e^x+1)^2}$ [duplicate]

I came across this integral when helping some friends with a statistical mechanics assignment, Mathematica reports it as $\frac{\pi^2}{3}$. So far I have noticed that the integrand is an even ...
0
votes
1answer
38 views

Convert Integral Rectangular to Polar

How can convert this problem $$ \int_0^2 \int_x^\sqrt{8-x^2} \left(x^2+y^2\right)^{3/2} dydx $$ I convert limits and funtion to polar cordinates as follows: $$ \begin{split} r^2 &= ...
0
votes
2answers
36 views

Logistic differential equation problem

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide any explanations. I'm having trouble turning the ...
0
votes
1answer
77 views

If the integral of $|f_n|$ converges to zero, so does the integral of $f_ng$ for integrable $g$

Let's assume that the $f_n$ are integrable and all bounded with the same bound. also, that $\int|f_n|\to 0$ as $n\to\infty$ (a) Prove under these assumptions that $\int f_ng\to0$ for any $g\in ...
1
vote
5answers
56 views

Prove that $\frac{\pi^3}{48} \le \int_0^{\pi/2}\frac{x^2}{2-\sin(x)}\,dx \le \frac{\pi^3}{24}$

Is it possible to prove that $$\frac{\pi^3}{48} \le \int_0^{\pi/2}\frac{x^2}{2-\sin(x)}\,dx \le \frac{\pi^3}{24}$$ without evaluating the integral?
1
vote
1answer
21 views

Integrals with functions as bounds

How to calculate integral such as $$\int_{g(χ)}^{φ(χ)} f(s) \, ds$$ where $F'(s)=f(s)$ Integrals like this appear often in PDE's .I'd like to know the whole theory i mean if there is a formula how ...
6
votes
2answers
67 views

Area of greatest integer function

Question: Find the area enclosed by the function: $$\left\lfloor\frac{\left|3x + 4y\right|}{5}\right\rfloor + \left\lfloor\frac{\left|4x - 3y\right|}{5}\right\rfloor = 3$$ where ...
2
votes
3answers
42 views

Is this an acceptable trig-sub and reversion from trig at the answer?

Today I had to take the indefinite integral $\int x^3 \sqrt{x^2-1} \, dx$ My steps: $x=\sec\theta$, $dx=\sec\theta\tan\theta\, d\theta$ $\displaystyle \int \sec^3\theta \sqrt{\sec^2\theta - 1} \, ...
1
vote
4answers
60 views

Evaluating integral with $e^{\sin x}$

I had this integral $ \int e^{\sin(x)} {\sin(2x)} dx$ I tried to split it up using integration by parts but I can't evaluate integral of $e^{\sin x}$
-2
votes
3answers
79 views

Is it possible to solve $\int_{0}^{1} t^4 \sqrt{1+t^2}\,dt$ [on hold]

How should I solve the below integral?$$\int_{0}^{1} t^4 \sqrt{1+t^2}\,dt$$
3
votes
1answer
34 views

How can I evaluate this contour integral?

Suppose we have the following contour integral, in the complex plane: $$ \int_{\gamma} \frac{e^{\frac{1}{z}}}{z^{2}} \; dz $$ where $\gamma (t) = e^{it}$ for $0 \leq t \leq 2 \pi$. To solve this, I ...
-1
votes
0answers
17 views

Integration of exponential and trigonometry [duplicate]

Please explain howto integrate this formula. Question Regards!
4
votes
2answers
47 views

Possible alternative for finding the Area under the floor function (aka, the integral of floor(x))

So, I had to ask myself the question as to what the area under the floor function could possibly be. I started by graphing the basic $\mbox{floor}(x)$ function (I personally use desmos.com for a nice ...
1
vote
1answer
54 views

An integral involving hyperbolic functions

$$ \large \displaystyle \int_0^\infty {\dfrac{e^{-2x} \tanh\frac{x}{2}}{x \cosh x}dx} = 2 \ln \frac{\pi}{2\sqrt{2}} $$ How to prove the above integral? What I tried : $\displaystyle I(s) = ...
0
votes
0answers
38 views

Integration by parts thrice [on hold]

To find real fourier coefficients this resolves to a triple integration by parts. When I let $x=u$ and $dv=e^xcosxdx$ to find the $cos(nx)$ coefficients, This translates to $u∫dv-∫∫dvdu$, My ...
2
votes
1answer
35 views

Integrating $ \frac{\int_{-\pi}^{\pi} \cos^2(x)|\cos(x)| |\sin(x)| dx }{\int_{-\pi}^{\pi} \cos^2(x)|\cos(x)| dx } $

I'm trying to integrate $ \frac{\int_{-\pi}^{\pi} \cos^2(x)|\cos(x)| |\sin(x)| dx }{\int_{-\pi}^{\pi} \cos^2(x)|\cos(x)| dx } $ I understand that $|\cos(x)|$ and $|\sin(x)|$ when integrated over $- ...
0
votes
0answers
17 views

Proof of the finiteness of integral (in option pricing)

I would like to ask for help with proving the finiteness of the following double integral. $$\int_{0}^{\infty}e^{\alpha+k}\int_{k+\zeta}^{\infty} (e^{-\zeta+x}-e^k)f(x)\ \mbox{d}x\ \mbox{d}k,$$ ...
0
votes
0answers
45 views

Show that, $\int_0^1\frac{1}{1+x^2}\sum_{n=1}^{\infty}x^{2^n-1}dx =\frac{2}{\pi}$ [on hold]

Show that, $$\int_0^1\frac{1}{1+x^2}\sum_{n=1}^{\infty}x^{2^n-1}dx =\frac{2}{\pi}$$
0
votes
1answer
27 views

Solving an integral by Cauchy Formula

I want to solve the integral $$\oint_{|z|=\frac{1}{2}}{\frac{e^{1-z}}{z^3(1-z)}dz}$$ Its a long time ago that I solved such integrals. Is it just by definition of the line integral? Maybe someone can ...
0
votes
2answers
82 views

Integral of $\frac{1}{( x^{2015} - x)}$

I am trying to find the integral of $\dfrac{1}{( x^{2015} - x)}$. Does anyone know how to do this? I can't possibly do a you substitutions right? Can't do partial fraction either.
2
votes
3answers
69 views

Determine the derivative $\frac{dy}{dx}$ of the integral

Determine the derivative of the integral $$ \,\int_{\sqrt x}^{0}\sin (t^2)dt $$ What does this question mean. I do not understand it and I think you can't integrate $\sin t^2\,$.
1
vote
0answers
21 views

Evaluate the integral $\iint \operatorname{curl}(yi+2j)\cdot n \, d\sigma $

Evaluate the integral $\iint \operatorname{curl}(yi+2j)\cdot n \, d\sigma $ where $\sigma$ is the surface in the first octant made up of part of the plane $2x+3y+4z=12$ and triangular in the ...
0
votes
0answers
45 views

Find all possible values of the integral

Find all possible values of $\displaystyle I= \int_C \frac{dz}{1+z^2}$, where $C$ is a curve with initial point $0$ and final point $1$ that does not meet the poles of $\dfrac{1} {1+z^2}$. It looks ...
0
votes
1answer
16 views

Factorising Iterated Integrals

I've been doing a bunch of homework about iterated double and triple integrals recently and a bunch of solutions seem to skip steps and use shortcuts that everyone seems to know, but no one seems to ...
0
votes
0answers
35 views

Seeking help with finding the general solution of this differential equation

I am trying to find the general solution of the following equation. $$\int_0^\infty \frac{\partial f(x, t)} {\partial t} \sin(x \xi) \, dx = \xi \int_0^\infty f(x, t) \cos(x \xi) \, dx -\alpha \xi ^2 ...
0
votes
0answers
46 views

Find the points on the curve $y=x+e^x$ at which the tangent line is horizontal.

Find the points on the curve $y=x+e^x$ at which the tangent line is horizontal. The answer was $(0,1)$, but I don't get it. I tried to take the derivative of the function and equal it to $0$ ...
-3
votes
2answers
31 views

Triple Integral with spherical polar coordianates [on hold]

By changing into spherical coordinates (or by any other method) evaluate the triple integral $$\iiint_V xyz \ dxdydz,$$ where $V$ is the volume in $\mathbb{R}^3$ deifned by the inequalities ...
4
votes
0answers
71 views

A generalization of an integral related with $\zeta(2)$

It is pretty well-known (and not difficult to prove) that: $$ \int_{0}^{+\infty}\frac{x}{e^{x}-1}\,dx = \zeta(2) = \sum_{n\geq 1}\frac{1}{n^2} \tag{1}$$ but what is known about $$ I_2 = ...
1
vote
2answers
37 views

Specific example of integrating a 1-form over a curve

I was given the following definition in my course but no corresponding examples: Supppose $\gamma:[a,b]\rightarrow{M}$ is a smooth curve and $\omega$ a 1-form on $M$ (so $\omega:M\rightarrow{T^*M}$). ...
3
votes
3answers
90 views

Prove that $2\int_0^\infty \frac{e^x-x-1}{x(e^{2x}-1)}dx =\ln(\pi)-\gamma $

I have to prove that given $\gamma=0.577216\ldots$, the Euler-Mascheroni constant, and $\pi=3.14159\ldots$, we have: $$2\int_0^\infty \frac{e^x-x-1}{x(e^{2x}-1)}dx =\ln(\pi)-\gamma $$
1
vote
0answers
23 views

Double integral over an annulus

Question: Let $D$ be part of the annulus $1\le x^2+y^2 \le 4$ lying in the first quarter of the $oxy$ plane where $x \ge 0, y \ge 0$ and below the line $y=x$ Evaluate the integral $$\iint_D\ ...
3
votes
1answer
57 views

Part of proof of the set of continuous integrable functions is dense in $L^1(\Bbb R)$

I want to prove: If $g$ belongs to $L(\Bbb R, \Bbb B, \lambda)$ and $\epsilon\gt 0$, then there exists a continuous function $f$ such that $\Vert g-f\Vert_1=\int \lvert g-f\rvert \,\text{d}\lambda \lt ...
0
votes
0answers
15 views

How to numerically evaluate a integral whose limits are functions of x (using Gauss quadrature rule)?

I am trying to numerically evaluate an integral $\int_q^1 \ln (\sum_i \alpha_ix_i) dq$, in which $\ln (\sum_i \alpha_i x_i)$ is related to $q$ via the following: $z_i=(1-q)\frac{\alpha_ix_i}{\ln ...
2
votes
2answers
36 views

evaluating $\int_0^\infty \int_y^\infty y^2e^{-x^4} \ dx \ dy$

evaluating $\int_0^\infty \int_y^\infty y^2e^{-x^4} \ dx \ dy$ my book states $$\int_0^\infty \int_y^\infty y^2e^{-x^4} \ dx \ dy = \int_0^\infty \int_0^x y^2e^{-x^4} \ dy \ dx$$ Could someone ...
1
vote
1answer
40 views

Natural Logs and Anit-Derivatives are kicking me

I am given a problem involving rates of flow, $F(t)=\frac{t+7}{2+t}$ is the rate at which a bucket is being filled. The same bucket is being emptied at a rate given by $E(t)=\frac{\ln(t+4)}{t+2}$. My ...
0
votes
0answers
19 views

Extending the Riemann integral to any compact set

One basically defines a Riemann integral on a closed interval. I'd like to extend the Riemann integral to any compact set. Let $K \in \Bbb R$ be compact. Let $f\colon K \rightarrow \Bbb R$, with the ...
0
votes
1answer
24 views

$0 \leq Y \leq M$ random variable, $p > 1$. Calculate $\mathbb{E}(Y^p)$

$0 \leq Y \leq M$ random variable, $p > 1$. Show that $\mathbb{E}[Y^p] = \int_0^M py^{p-1}\mathbb{P}[Y \geq y] dy$ My attempt: $\mathbb{E}[Y^p] = \int_0^{\infty} Y d\mathbb{P} = \int_0^{M} Y ...
1
vote
1answer
35 views

How to evaluate $\lim_{c \rightarrow \infty} \int_{-c}^c f(x)dx$

I'm trying to evaluate: $$\lim_{c \rightarrow \infty} \int_{-c}^c \frac{1+x}{1+x^2}dx$$ but I don't understand how to evaluate $$\lim_{c \rightarrow \infty} \int_{-c}^c f(x)dx$$ How?
0
votes
0answers
11 views

Computing the normal vector when finding the flux

Use a parametrization to find the flux $\int \int_S F\cdot n$ $d\sigma$ across the surface in the given direction: $F=xy\overrightarrow i -z\overrightarrow k$ outward (normal away from the z-axis) ...
2
votes
1answer
33 views

Integral analog of geometric series

We all know that $$ \frac{1}{1-z}=\sum_{m=0}^\infty z^m\ , $$ for $|z|<1$. The challenge I would like to pose is: find (possibly as simple and elegant) integral representations (as many as you can) ...
0
votes
0answers
17 views

Parametrising a side of a cuboid

Question: Suppose the surface S is bounded by 6 planes $$x=0,x=2,y=0,y=4,z=0,z=1$$ Parametrise two of the surfaces. My attempt: S0 I picked the "floor" face of the cuboid i.e. $x= 0, x=2, y=0,y=4, ...
0
votes
0answers
18 views

Stoke's Theorem?

Let $S$ be the portion of the plane $x+y+z=1$ that lies in the first octant, and let C be the boundary of S, traversed counterclockwise. Calculate $\int_{C} F.dr$ where ...
0
votes
1answer
14 views

Use the cylindrical shell method to find the volume

Okay so here i have a question: Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. $y = 5x^2$ , $y = 30x ...
1
vote
1answer
24 views

Solving a surface integral using Gauss' Divergence Theorem

Question: Use Gauss's theorem to solve $$\iint_S F\cdot n~dS$$ given $$F(x,y,z)=(x,xy,z)$$ where S is the surface $$x^2+y^2= z^2, z \in [0,1]$$ My attempt: I have the solution and method for the ...
1
vote
2answers
55 views

How to integrate the following? $\int_{0}^{+\infty}\frac{1-\cos x}{x^{\alpha+1}}\,dx$

$$2\alpha\int_{0}^{\infty}\frac{1-\cos{x}}{x^{\alpha+1}}dx=?$$ I know that it should be solved by integrating on a contour of two semicircles with radius $\epsilon$ and $T$, and the real line. ...
0
votes
1answer
40 views

The limit of $\lim_{\Delta{x}\to0}\sum{_0^\infty}(2x\Delta{x})$

I was trying to derive integrals of some elementary function through summation notation. It was-- $\lim_{\Delta{x}\to0}\sum{_0^\infty}(2x\Delta{x})$
1
vote
2answers
73 views

Integration equals another integration

$\displaystyle\int f(x)\ dx=\displaystyle\int g(x)\ dx$ So what is the relation between $f$ and $g$ I found this solution but i am not sure it is right or not : $\displaystyle\int (f(x)-g(x))\ dx=0$. ...
0
votes
1answer
20 views

An integral related to the harmonic number

Definitions $\Gamma(s)$ is a Gamma function Defined as $\Gamma(s+1)=s!$ $H_{n,s}=\sum_{k=1}^{n}\frac{1}{n^s}$ We proposed: proof that, $$-\frac{1}{\Gamma(s)}\int_0^\infty x^{s-1}\cdot ...
0
votes
1answer
15 views

Problem with multiple integrals and integration limits

I'm working in this integration. $\int_{1}^{2}\int_{1/Y}^{y}\sqrt{\frac{y}{x}}\left(e^{\sqrt{xy}}\right)dxdy$ I make this: $u=\sqrt{xy}$ ,$v=\sqrt{\frac{y}{x}}$, $x=\frac{u}{v}$, $y=uv$ For ...
1
vote
1answer
43 views

Gaussian integral $\int_{-\infty}^\infty \exp(-(x+\mathrm iY)^2)\,\mathrm d x$ along $[-R,R]+\mathrm i[0,Y]$

Use integration along $\partial Q$ of $Q=[-R,R]+\mathrm i[0,Y]$ to show that for all $Y\geq 0$ it holds that $$\int_{-\infty}^\infty \exp(-(x+\mathrm iY)^2)~\mathrm dx = \int_{-\infty}^\infty ...