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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1answer
74 views

Integral $\int_{0}^{1} \frac{x^{2n +1}}{\sqrt{1 - x^2}} dx$ with n

Find a general expression for $$\int_{0}^{1} \frac{x^{2n +1}}{\sqrt{1 - x^2}} dx$$ for every natural n. Is there any common algorythm for such integrals?
0
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1answer
30 views

definition of $L^p$ space on the boundary

Let $D$ be a bounded open set in the complex plane $\mathbb{C}$. (definition) $f\in L^1(\partial D)$ if and only if $\int_a^b |f(\gamma(t))||\gamma'(t)|dt$ is finite. (where $\gamma$ is the ...
5
votes
1answer
100 views

Continuous-time version of Fatou's lemma

I have just read a textbook on stochastic processes that implicitly uses the fact that \begin{equation} \int \liminf_{t \to \infty} f_t \leq \liminf_{t \to \infty} \int f_t, \end{equation} for ...
0
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1answer
42 views

Show $\mathcal{L}\left\{\frac{1}{t}f(t)\right\} = \int_{s}^{\infty}F(u)du$ [duplicate]

Show for $\mathcal{L}$, the Laplace transform, that $$\mathcal{L}\left\{\frac{1}{t}f(t)\right\} = \int_{s}^{\infty}F(u)du.$$ I know that $\mathcal{L}\left\{ t^n f(t) \right \} = (-1)^n ...
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5answers
196 views

Evaluate $\int\limits_0^{\pi} \frac{dx}{1+2\sin^2x}$

After making u substitution two times, I am getting indefinite integral as $$\int\limits\dfrac{dx}{1+2\sin^2x} = \dfrac{\arctan(\sqrt{3}\tan(x))}{\sqrt{3}}+ C$$ I am stuck at working the bounds ...
0
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1answer
22 views

Last step of making integral stationary

In a problem making an integral stationary, I'm being told that this last implication is wrong, why? Is my integration wrong? $$ \frac{d}{dx} y(x) = \pm\frac{c_1x^2}{\sqrt{x^2 - c_1^2}}$$ $$\implies ...
3
votes
5answers
74 views

Integrate $\int \cos^4 xdx$

$$\int \cos^4 xdx = \int (1 - \sin^2x)^2 $$ I tried using $\cos^2(x) + \sin^2(x)$ = 1. This was on the integration by parts section of my textbook. The integral I came out with is given me a hard ...
3
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4answers
137 views

Calculation of $\displaystyle \int\frac{1}{\tan \frac{x}{2}+1}dx$

Calculation of $\displaystyle \int\frac{1}{\tan \frac{x}{2}+1}dx$ $\bf{My\; Try}::$ Let $\displaystyle I = \displaystyle \int\frac{1}{\tan \frac{x}{2}+1}dx$, Now let $\displaystyle \tan ...
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0answers
50 views

Proving inequality with complex Riemann-Stieltjes Integral

I am trying to prove the following proposition (IV.1.17b) from Conway's Functions of One Complex Variable I: Let $\gamma$ be a rectifiable curve and suppose that $f$ is a function continuous on $\{ ...
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1answer
38 views

Having trouble proving this integral is infinite

I am working on an assignment and I have to prove that $$\frac{x^2-y^2}{(x^2+y^2)^2} \notin L_1\mathbb{R}^2)$$ to justify why Fubini's Theorem does not apply. I figured that the best way to do this ...
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2answers
79 views

Integration Question: Completing the Square/Trig Sub yields a different answer than integral table.

After the completing the square, $$\int \frac{dx}{x^2 + 2x - 3}$$ becomes, $$ \int \frac{dx}{(x+1)^2 - 4}$$ The integral table in my book says the antiderivative is, $$\frac{1}{2a} ln \, ...
23
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1answer
542 views

Prove that $\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx =\frac{\pi e}{24} $

I've found here the following integral. $$I = \int_{0}^{1}\sin{(\pi (1-x))}x^x(1-x)^{1-x}\,dx=\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx=\frac{\pi e}{24}$$ I've never seen it before and I also ...
0
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3answers
43 views

Integrating $\cos(x)^3dx$

My attempt at integrating $\cos(x)^3dx$: $$\begin{align}\;\int \cos^3x\mathrm{d}x &= \int \cos^2x \cos x \mathrm{d}x \\&= \int(1 - \sin^2 x) \cos x \mathrm{d}x \\&= \int \cos x dx - \int ...
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1answer
19 views

Integration by substitution - determine support

I was wondering if anyone may be able to direct me to the correct materials to understand how the support of a function changes with substitution. For example, if I have a simple integral with a ...
1
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1answer
38 views

Compute volume of tetrahedron using a triple integral

I'm trying to compute the volume of a tetrahedron with the vertices (0, 0, 0), (0, 0, 1), (2, 0, 0), (0, 2, 0). It needs to be done using a triple integral. Not allowed to use "det" or other ...
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1answer
58 views

Indefinite integral $\int t \cdot \cos^3(t^2)dt$

I am having trouble integrating $$\int t \cdot \cos^3(t^2)dt$$ Progress I have made $u=t^2$ which makes the problem $1/2 \int \cos^3(u) du$. After writing that out I subsituted $v=\sin(u)$ ...
2
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1answer
67 views

Evaluating $\;\int x\cos(5x-1)\,\mathrm{d}x$

Integration by parts: 1) let $u = x$, $du = dx$, $v = \frac{1}{5} \sin (5x- 1)$, $dv = \cos (5x -1) dx$ $udv = x \frac{1}{5} \sin x ( 5x- 1) - \frac{1}{5} \int \sin (5x-1) dx$. 2) let $u = \sin ...
1
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2answers
71 views

Explain how to get the right solution of y $dy/dx=y$

When solving the following equation to find y as a function of x: \begin{equation} dy/dx=y \end{equation} First I divide both sides by $y$ and multiply both sides by $dx$: $dy/y=dx$ Then I ...
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1answer
44 views

Prove that $\int\limits_{-a}^{a}f(x)dx = 2 \int\limits_{0}^{a}f(x)dx$ if function $f(x)$ is even

Prove that $\int\limits_{-a}^{a}f(x)dx = 2 \int\limits_{0}^{a}f(x)dx$ if functio is even. $\int\limits_{-a}^{a}f(x)dx = \int\limits_{-a}^{0}f(x)dx + \int\limits_{0}^{a}f(x)dx$ ...
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1answer
85 views

Use the joint probability density to find $P(X+Y>3)$

$$ f(x,y) = \begin{cases} e^{-x-y}, & \text{for x>0 and y>0} \\ 0, & \text{elsewhere} \end{cases} $$ I found this joint probability density by solving a previous problem that gave me ...
3
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2answers
56 views

Solving $\int_{-\infty}^{+\infty}e^{-2\alpha|x|}\cos^2(x)\,dx$

I'd like some help solving the integral $$ \int_{-\infty}^{\infty} e^{-2 \, \alpha \, |x|} \cdot \cos^2(x) \; \, dx $$ with $\alpha > 0$ I just assumed 'integration-by-parts' was the way to go, ...
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0answers
21 views

Analytical calculation of mutual information of a sine time series

Suppose I have the following time series $$ x_t=\sin(0.02\pi t) $$ and I want to calculate the mutual information $I_{x_t,x_{t-\tau}}$ which by definition is $$ ...
0
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0answers
31 views

Interpolating array of integrals

I have an array of integrals, that I want to interpolate and differentiate to get result array. What kind of interpolation should I use to get a reasonably smooth output? (e.g. the output is ...
2
votes
2answers
43 views

how can I derive this identity involving the integral representation of Digamma function

In here, there is an identity in equation 17 $$ k {N-1\choose{k}}\int_0^1 p^{k-1}(1-p)^{N-k-1}\log p\,dp =\psi(k)-\psi(N), $$ where $N$, $k(<N)$ are integers and $p$ can be regarded as the ...
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0answers
31 views

Quaternion Integration - And conversion to 3D matrix

I have a rotation matrix let us say $R(t)$ and its quaternion $q(t)$. We know already how to convert a quaternion to rotation matrix. Now if I want find $\int R(t) \ dt \tag1 $ can we do that in ...
2
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1answer
84 views

Volume of sphere with triple integral

Using the same notations as in this picture : The element of volume is: $r^2 \sin(\theta) \, dr \, d\theta \, d\phi$ If I try to create the volume visually, I begin with integrating $r$ between ...
2
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3answers
84 views

If a function integrates to zero against every even function, then it is odd

I'm in the process of figuring this one out as well: Let $f : [−1, 1] \to\mathbb R$ be a smooth function. Show that if $$ \int_{0}^{1}f(x)v(x)dx = 0 $$ for every smooth function $v : [0, 1] \to ...
11
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1answer
686 views

A Putnam Integral $\int_2^4 \frac{\sqrt{\ln(9-x)}\,dx}{\sqrt{\ln(9-x)} + \sqrt{\ln(x+3)}}.$

This is a Putnam Problem that I have been trying to solve (on and off) for two years, but I have failed. I am in Calculus BC. This problem comes from the book "Calculus Eighth Edition by Larson, ...
0
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2answers
107 views

If a function integrates to zero against every function with zero mean, then it is constant

Struggling with this calculus problem: Let $f : [0, 1] \to \mathbb R$ be a smooth function. Show that if $$ \int_{0}^{1}f(x)v(x)dx = 0 $$ for every smooth function $v : [0, 1] \to\mathbb R$ for ...
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4answers
86 views

Integrate $(1+7x)^{1/3}$ from $0$ to $1$

Integrate $(1+7x)^{1/3}$ from $0$ to $1$ So using substitution, I'm able to get to $$\frac17 \frac34 u ^{4/3}$$ Pretty sure that part is right, but I'm getting stuck after that.
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0answers
109 views

Fourier transform of squared exponential integral $\operatorname{Ei}^2(-|x|)$

Let $\operatorname{Ei}(x)$ denote the exponential integral: $$\operatorname{Ei}(x)=-\int_{-x}^\infty\frac{e^{-t}}tdt.$$ Now consider the function $\operatorname{Ei}(-|x|)$. ...
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5answers
2k views

Integrate cos (lnx) dx

Integrating by parts: I'm having a hard time choosing the $u$, $du$, $v$ and $dv$... I gave it a shot. $u = \ln x \implies du = 1/x \ dx$ $v= \ ?$ $dv = \cos \ dx$
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0answers
90 views

Application of integrating $\cos^4 x$?

A student asked a colleague the other day for a practical application that involved needing to integrate the fourth power of cosine, but no one here could think of one off-hand other than some volume ...
0
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1answer
33 views

Differentiating a function and using the result to calculate the indefinite integral of another.

We should differentiate the function $f(x) = \sqrt{cosx}$ and use the result to calculate the indefinite integral $\int \frac{sinx}{\sqrt{cosx}}dx$. So I started by differentiating $f(x) = ...
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1answer
291 views

Product of two exponentially distributed random variables

I am trying to find the close form expression of probability distribution of $Z$ such as $Z=X_1X_2$ where $X_1$ and $X_2$ are two independent exponentially distributed variables with PDF ...
2
votes
0answers
48 views

An upper Bound for $(f(a))^2$, $a\in[0,1]$ in terms of $\int_0^1(f(x))^2dx$

Is there any way to find an upper Bound for $(f(a))^2$, $a\in[0,1]$ in terms of $\int_0^1(f(x))^2dx$. There is a commonly used upper bound in terms of $\int_0^1(f_x(x))^2dx$, but I do want to make ...
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0answers
38 views

Integration by Parts in matrices

Given Data in the question We have a given equation based on matrices as follows $\frac{\mathrm{d} R(s)_{3\times3}}{\mathrm{d} s}=R(s)_{3\times3}K(s)_{3\times3} \tag 1$ $\frac{\mathrm{d} ...
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3answers
38 views

$x e ^{-\frac{x^2}{2}}$ at $\infty$ to $-\infty$

I want to know how to explain $\left. \left(x e ^{-\frac{x^2}{2}} \right) \right|_{- \infty} ^{\infty}$ is zero? Is it because the speed of exponentiation is greater than that of linear? How to ...
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0answers
30 views

multivariate quadrature

Assume that $f:\mathbb{R}^n\ \to \mathbb{R} $. We want to approximate the integral, $\int_{I_d} f \, d\mu$. Let $U^{m_i}$ be a quadrature rule in $x_i$ in direction of $x = (x_1 , \dots , x_n)$, with ...
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2answers
25 views

What would the next step be in this integral problem?

I have to integrate the following $\int \frac {3x^2+x+5}{3x^2+x+4}dx$ what I can see is that i can substitute $3x^2+x+4$ for $u$ thus making my integral $\int \frac {u+1}{u}du$ this will give me $\int ...
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0answers
44 views

how we integrate function s respect to y?

I would like to integral this function $$ \int\exp\big(x^2-y^2\big) \Big( \!2y\cos(2xy) +2x\sin(2xy)\Big)\mathrm{d}y $$ Thank you! $\exp\big(x^2-y^2\big)$ is a common factor for sin and cos
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1answer
66 views

Evaluating $\int x^n e^{x}dx$

I consider, for $n=0,1,2,...$, $$ u_n(x)=\int x^n e^{x}dx.$$ I've performed an integration by parts giving $$ u_n(x)=nx^{n-1} e^{x}-nu_{n-1}(x).$$ I'm looking for a closed form. Thank you.
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0answers
37 views

Derive Runge-Kutta matrix with known weights and nodes

Can I derive a Runge-Kutta method by choosing freely the weights and the nodes? what are my constraints? So, if this is the general form of the explicit RK method: $$ y_{n+1} = y_n + \sum_{i=1}^s b_i ...
0
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0answers
26 views

Find function that gives Fourier transform value of 1

I am trying to find a function $f(a)$ so that the following expression $$ f(a) e^{-\frac{1}{2}\frac{x^2 + y^2}{a^2}} $$ has a Fourier transform equal to 1 as $a \rightarrow 0$. The reason I am doing ...
0
votes
0answers
45 views

The partial derivatives of the function $s=\int _u^v\frac{(1-e^t)}{t}dt$

If $s=\int _u^v\frac{\left(1-e^t\right)}{t}dt$, I want to find $\frac{∂s}{∂v}$ and $\frac{∂s}{∂u}$ and their limits as $u$ and $v$ tend to zero. first i find for $\frac{∂s}{∂v}=\frac{∂}{∂v}\int ...
3
votes
4answers
57 views

What do limits of functions of the form $te^t$ have to do with l'Hopital's rule?

I have an improper function that I have to integrate from some number to infinity. Once integration is done, the function is of the form $te^t$. What I'm wondering is what does this have to do with ...
1
vote
1answer
73 views

Green-Riemann Theorem

I calculated the circulation of the vector field : $$\vec{v} = -y\omega \, \vec{i} + x\omega \, \vec{j}$$ over the ellipse : $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ I found $2 \pi \omega a b$. ...
0
votes
1answer
28 views

On the equivalency of two indefinite integrals using u substitution.

I am reading the Separation of variables page on wikipedia, at a certain point it states that the following equation Is equal to (1) because of the substitution rule of integrals. The ...
1
vote
1answer
51 views

Help explaining divergence theorem example

I am looking at an application of the divergence theorem, and I don't understand what's going on. Could anyone explain how to go from the first expression to the second expression (which can then be ...
5
votes
1answer
39 views

Integral versus hypergeometric series: how to solve this?

How can I resolve the following indefinite integral using hypergeometric series? $$ \int (x^3 + 1)^\frac{1}{3} \,dx $$ Wolfram Alpha indicates that the series of Appell are used, but how to get to ...