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
36 views

Mass of a sphere of radius 1. [on hold]

A sphere of radius 1 is given. Using spherical coordinates, solve for the Mass. Density varies directly with the square of the distance from the center of the ball.
1
vote
0answers
57 views

How to calculate this integral?

I got confused with this Markov Chain problem: suppose the kernel $Q$ is $Q_x=N(cx,1)$, $c$ is a fixed constant with $|c|<1$ and the stationary distribution is $\pi=N(0,\frac{1}{1-c^2})$. I want ...
0
votes
0answers
28 views

Computing the integral

Let $A=(a_{ij})_{1 \leq i,j \leq n}$ be a positive definite and symmetric matrix of dimension $n \times n$. I want to compute $$\int_{\mathbb{R}^n} \exp( -\frac{1}{2} \sum_{1 \leq i,j \leq ...
-2
votes
2answers
55 views

$\int _{k\pi }^{\left(k+1\right)\pi }\:\left|\sin\left(x\right)\right|dx$ [on hold]

How can I solve the following integral? $\int _{k\pi }^{\left(k+1\right)\pi }\:\left|\sin\left(x\right)\right|dx$
0
votes
1answer
22 views

When should I recalculate starting values for my physics engine's multistep integration?

I'm writing a physics engine that uses an Adams-Bashforth and Adams-Moulton predictor-corrector pair for integration, and Runge-Kutta 4 to calculate starting values for their initialization. I am ...
4
votes
1answer
79 views

Can $\oint_{|z|=2}z^3 \bar {z} e^\frac{1}{(z-1)} dz$ be solved?

How we can calculate the result of following Integral? $$\oint_{|z|=2}z^3 \bar {z} e^\frac{1}{z-1} \mathrm{d}z$$
0
votes
1answer
54 views

Integrating $\int x \sin \sqrt x dx$ and substituting $u=\sqrt x$ [duplicate]

Is it wrong to substitute $u=\sqrt x$ when integrating? Here's what I mean, I have to integrate: $\int x \sin \sqrt x dx$. I defined $u=\sqrt x$ so it's: $$\int u^2\sin u ...
2
votes
0answers
14 views

An extension of change of variables in double (and $n$-?) integrals - second-order Jacobian?

I'm aware that there are many, many questions regarding changing variables in double and triple integrals. The equation that typically pops up in textbooks is \begin{align} ...
1
vote
1answer
24 views

existence of an improper integral

Let $f: [1, \infty ) \to \mathbb{C}$ be a continuous function with a bounded antiderivative $F(x)$ on $[1, \infty)$. Show, that the integral $$ \int_1^\infty \frac{f(x)}{x^s} dx$$ exists for each $s ...
1
vote
3answers
319 views

Solve for velocity, if acceleration is a function of velocity?

Yes, this is a canned question, because canned questions are simply solvable to understand ideas. No, this is not homework. For simplicity, let's assign values. A given object weights 1000 kg, it ...
0
votes
1answer
20 views

Leibniz rule for an improper integral

It follows from leibniz rule that if $\frac{\partial f}{\partial \theta_0}(\theta,\theta_0)$ exists then $$\frac{d}{d\theta_0}\bigg(\int_0^{\theta_0}f(\theta,\theta_0)d\theta\bigg)=\int ...
0
votes
0answers
11 views

Recursive relationship for Peano Baker Series

The Peano Baker Series is a integral has the following form $$\varPhi(h,0)=I+\intop_0^h G(t_{1}) \, dt_1 + \intop_0^h G(t_1) \intop_0^{t_{1}} G(t_2) \, dt_2 \, dt_1 + \intop_0^h G(t_1) ...
0
votes
0answers
37 views

Fourier transform of a tough composite function (sinc, sqrt, polynomial…)

Is it possible to compute the Fourier transform of $\mathrm{sinc}(\sqrt{1+x^4})$ in closed form? It appears the problem to be suited for contour integration, and I started to tackle the mere ...
0
votes
1answer
29 views

How to visualize a line integral

I was studying for multivariable calculus and I came across the line integral section. Visually, I perfectly understand why $\int_{t_1}^{t_2} f(x(t),y(t))s'(t)\,dt$ computes the area under $f(x,y)$ ...
0
votes
0answers
36 views

Calculate the Volume of the unitary n-ball

To do this first I need to prove that : $$\displaystyle \int_R \sin^{n-2}\phi_1 \sin^{n-3}\phi_2\cdots\sin \phi_{n-2} d\theta d\phi_1 \cdots d\phi_{n-2} = \frac{2\pi^{n/2}}{\Gamma(n/2)}$$ where $ ...
2
votes
0answers
64 views
+50

Does $\operatorname{div}\left(\nabla G +xG\right)=0\Longleftrightarrow \nabla G +xG=0$?

Let $G$ be a function of $\mathcal{C}^2(\textbf{R}^d$;$\textbf{R}^*_+)$ such that $G \in \operatorname{L}^1(\textbf{R}^d)$. I read on the Internet that one has the following equivalence ...
3
votes
1answer
68 views

Evaluating: $I_1 = \sin^{-1} \left(\sqrt{\frac{x}{x+a}}\;\right) $

$$I_1 =\int \sin^{-1} \left(\sqrt{\frac{x}{x+a}}\;\right) dx= ?$$ I tried substitution: $\sin^{-1} \left(\sqrt{\frac{x}{x+a}}\;\right) = \Xi$, but then I'm not able to do anything after the resulting ...
0
votes
1answer
24 views

Line integral over vector field $F(x,y) = (\sin(x) + 3x^2,2x-e^{-y^{2}})$

I have to calculate line integral over vector field $F(x,y) = (\sin(x) + 3x^2,2x-e^{-y^{2}})$. The path is the unit circle edge, from (1,0) to (-1,0). So I have ...
8
votes
0answers
63 views

The inverse Laplace transform of $ s^{3/2}-a-bs \over s^{3/2}+a+bs$

How can I solve the inverse Laplace transform as below: $$\mathscr{L}^{-1}\left( s^{3/2}-a-bs \over s^{3/2}+a+bs \right) $$ where a and b are constants. Hint: we can consider $${ s^{3/2}-a-bs ...
2
votes
1answer
61 views

substitution integration question

I want to integrate $\int \sqrt{1 - x^2} dx $. When I substitute $x = \sin θ$ , I get the right answer. ( $ \cos^2\theta$ before integration) But when I substitute $x = \cos θ$ , I don't get the ...
-1
votes
1answer
48 views

Nobody can help me ? I can't believe that…

Okay, we have $I_n=\int _{\pi }^{2\pi }\:\frac{\left|sin\left(nx\right)\right|}{x}$, and we need to prove that: 1)$I_n\le log\left(2\right)$ $,\:\:\:\:\:$ why just log(2) ? can not be 1? ...
0
votes
1answer
49 views

Why do we treat differential notation as a fraction in u-substitution method [duplicate]

How did we come to know that treating the differential notation as a fraction will help us in finding the integral. And how do we know about its validity? How can $\frac{dy}{dx}$ be treated as a ...
-2
votes
1answer
56 views

integration of $e^{-\frac{t}{\tau}}$ [on hold]

$$\int_0^\tau e^{-\frac{t}{\tau}}\mathrm{dt}$$ Please give a very detailed explanation. Anyway, the answer to this is $\tau \left(1-\frac{1}{e}\right).$
-2
votes
0answers
42 views

wavepacket in one-dimensional quantum mechanics [on hold]

the problem is in the picture. The problem is about fourier series, but I do not see how it is related.
1
vote
2answers
44 views

Help in checking work to volume problem

Alright, I've got plenty of work to show for this one. The question goes... Suppose $ D$ is a solid bounded below by the sphere $ x^2 + y^2 +(z-1)^2 = 1$ and above by the cone $ z = \sqrt{x^2+y^2} $ ...
-4
votes
0answers
33 views

How to integrate complex numbers? [on hold]

Complex numbers have 2 variable so does it's integration entail contour integration or can we integrate assuming one variable to be a constant in terms of the other or do we try to find a relation ...
1
vote
1answer
31 views

Hard Integral [Heat Equation + Fourier Sine Series]

I encountered this integral while doing a heat equation problem in Advanced Calculus. How does the person evaluate the integral involving $$\int_0^\pi \sin x \cos (nx) \: dx $$ Can someone ...
2
votes
1answer
43 views

Help with evaluating a double integral

How would I go about evaluating... $$\int_{0}^{2}\int_{0}^{4-x^2} \frac{xcos(3y)}{4-y}dydx$$ I don't really have any work to show, since I don't even know how to start this problem, but I'll offer ...
0
votes
1answer
34 views

Propose a change of variables to simplify the double integral of (y-2x)^2(x+y)^2dydx

$$\int_{-1}^1\int_{2x-2}^{2x+2}\ (y-2x)^2\ (x+y)^2\ dy\ dx$$ I began by proposing $u=y-2x$ and $v=x+y$ and then solving each for $x$ and $y$ then computing the Jacobian which came out to be $-1/3$. ...
0
votes
1answer
27 views

Congruent Sub-Intervals with Reimann-Integrable Functions

Let $f:[a,b]\to\Bbb R$ be a Riemann-integrable function. Prove that for each $\sigma\gt0$ there exists a partition $\mathcal P$ of $[a,b]$ into congruent sub-intervals(that is, $x_{j}=a+{j(b-a)\over ...
-1
votes
2answers
53 views

How to integrate $\frac{1}{(1+a\cos x)}$ from $-\pi$ to $\pi$ [duplicate]

How to solve the following integration?$$\int_{-\pi}^\pi\frac{1}{1+a \cos x}$$
0
votes
1answer
16 views

Evaluating $\int f_1 f_2$ for measurable functions $f_1,f_2\ge 0$

If $f_1,f_2$ are nonnegative measurable functions defined on a subset $A$ of $\mathbb{R}$, and $f_3(y)=\int_{\lbrace x\in A: f_2> y\rbrace } f_1(x)dx$. Then $$\int_A f_1(x)f_2(x)dx=\int _0 ^\infty ...
1
vote
2answers
26 views

How would you use Fubini's theorem to solve $\int_{0}^{1}\int_{\arcsin(y)}^{\frac{\pi}{2}} e^{\cos(x)} \, dx \, dy$

I think I need to re-define the upper and lower limits, but in any case I end up with: $\int e^{\cos(x)} \, dx$ which doesn't allow me to solve the equation.
1
vote
1answer
73 views

Which of the following is true for $\int_{1}^{0} x\ln x\, \text dx$?

Which of the following is true for $\int_{1}^{0} x\ln x\,\text dx$ it is equal to $−1/4$ it is divergent it is equal to an irrational number does not have a closed form it is impossible to ...
0
votes
1answer
18 views

Trapezoidal rule in 2 dimensions

I'm using trying to integrate a function in MATLAB using the trapezoidal rule. I'm struggling to get the limits right and how to set up the steps. The limits for $x$ are $[0,2]$ and the limits for ...
1
vote
3answers
51 views

Definite integral of function is zero

I am attempting to solve an equation wherein $\int_{-\infty}^\infty f(x) \, dx = 0$. There obviously exist some some solutions, such as $f(x) = xe^{-x^2}$ and trivially $f(x) = 0$, but is there a ...
-4
votes
4answers
78 views

How to integrate $\frac{x}{\sqrt{1+x^2}}$

I wonder how to approach this integral - $\frac{x}{\sqrt{1+x^2}}$ . I see that it is just $\sqrt{1+x^2}$ but can't find the way to get to there. I tried integration by parts, but I get ...
1
vote
1answer
46 views

integrating to find a mean

I am trying to find the mean of the function $\exp(-2r)$ in 2 dimensions. This should then be the equivalent of doing ...
0
votes
0answers
38 views

Seemingly straightforward: Integral of ratio of polynomials

So, I have $$ I=\int\frac{x (a-x)^{b}}{(c-\sqrt{d+g x+x^2})^{\frac{3}{2}(c-1)}}dx $$ where $a,b,c,d$ and $g$ are all constants and the integrand is always real and positive. Mathematica 10 and ...
2
votes
1answer
20 views

Expectation of product of two correlated gaussian variables

$\newcommand{\var}{\operatorname{var}}$It seems I can not find the answer anywhere, please point it out how to calculate. Here, I have $X$, $Y$,$G$,$X_D$ and $Y_D$,both are Gaussian variables, and ...
5
votes
1answer
57 views

show that $\int_0^{\infty}\ln(2x)\frac{ax-u(1-e^{-2ax})}{\sinh^2(ax)}x^{2u-1}dx=\frac{1}{2u}(\frac{\pi}{a})^{2u}|B_{2u}|:u \in N$

$$\int_0^{\infty}\ln(2x)\frac{ax-u(1-e^{-2ax})}{\sinh^2(ax)}x^{2u-1}dx=\frac{1}{2u}(\frac{\pi}{a})^{2u}|B_{2u}|:u \in N$$ using real or complexe analysis where $B_{2u}$ is bernoulli number
1
vote
0answers
44 views

Integration.Matrix.Determinant.Inverse.Trace.

Given $$ I_n=\int_0^1\frac{x^n}{x^{2012}-1}{\rm d}x\text{ and }J_n=\int_0^1\frac{x^n}{x^{2013}+1}{\rm d}x\quad\forall n>2012, n\in\mathbb N$$ If the matrix $$\rm A=[a_{ij}]_{3\times3}\text{ where ...
4
votes
1answer
70 views

Integral Inequality $\int\limits_0^1f^2(x)dx\geq12\left( \int\limits_0^1xf(x)dx\right)^2.$

Let $f:[0,1]\to\mathbb{R}$ be a continuous function such that $\int\limits_0^1f(x)dx=0$. Prove that $$\int\limits_0^1f^2(x)dx\geq12\left( \int\limits_0^1xf(x)dx\right)^2.$$ My approach as follow Let ...
3
votes
1answer
43 views

Improper Integral with trigonometric functions

Determine if the following integral converges: $$\int_{-\infty}^{\infty}\frac{\cos(x)}{x^3+4x}dx.$$ So far I've thought about using the comparison test but I'm not sure how to implement it. My first ...
1
vote
0answers
16 views

Region bounded (Positive x-axis and y-axis)

Curve A is defined as $\ln (x^2 + 4)$ Find the region bounded by curve A, positive X axis and Y axis. My attempts, I've drawn the function, split it into two parts and found out that it's the area ...
0
votes
1answer
70 views

Can anyone please help with this integral. Very much appreciated..

Now , i've tried a couple of different substitutions and integrating partially but unfortunately to no luck, was wondering on your thoughts on it. I'd also be very thankful if someone were to have a ...
0
votes
0answers
19 views

Verify Solution inhomogeneous differential equation.

i'm doing a problem that should be handed in tomorrow. One of the problems is a differential equation and i'm a bit vague(right word?) on these types of problems.I'll show all the steps i've done. We ...
1
vote
0answers
41 views

Prove that these result are the same

I did this trigonometric integral in two different ways, and the results that I got were with two different trigonometric functions, $\sec x$ and $\tan x$. The integral is: $\mathbf{\int tan^{5}x \, ...
1
vote
1answer
49 views

Taylor polynomial for an integral

This is the first time encountering a Taylor expansion along with an integral, so I am wondering how I should proceed. Question: $Consider \space the \space function$ $$F(x) = ...
2
votes
1answer
46 views

a question about the evaluation of integral [duplicate]

Let $\alpha:[0,1] \to R$ be the Cantor function. Evaluate $$\int_{0}^{1}xd\alpha $$and $$\int_{0}^{1}x^2d\alpha.$$ I know that the Cantor function is continuous and monotone increasing, how can I ...