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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18 views

Integration of $\frac{x^2}{2\left(e^x+1\right)}$

Let: $$f(x) = \int \frac{x^2}{2\left(e^x+1\right)}dx $$ Is there a way to find $f(x)$? I've tried through integration by parts, but that didn't work out. If substitution is the answer, I can't see ...
1
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1answer
42 views

Is there a nice closed form to $\int_0^{\pi/2} (\log \sin x)^n\text{ d}x$ for $n\in \Bbb{Q},n\gt 1$?

For $n\in \Bbb{N}$, $$\int_0^{\pi/2} (\log \sin x)^n\text{ d}x=\frac{1}{2^{n+1}}B^{(n)}\left(\frac{1}{2},\frac{1}{2}\right)$$ Can we extend that result a bit further, to $n\in \Bbb{Q},n\gt 1?$
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5answers
60 views

integration of $1/x$ a counterexample to the rule

We know that the integration of $\displaystyle\int\frac{1}{x}\,dx=\log\left(|x|\right)$+$c$ with $x\neq 0$ , but if we go by normal rule then it becomes $\infty$. Is this a counterexample to the rule ...
2
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1answer
37 views

Compute $\int_0^1 \frac{ 1}{1 + x^{1/2}}\,dx$.

Basically, the question is $$\int_0^1 \frac{1}{1+x^{1/2}}\,dx.$$ I have no idea how to approach this and have spent hours to no avail. Any help would be gladly appreciated. Thanks!!
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3answers
40 views

what's wrong with this integral by parts calculation?

Today, I just finda confusing quesiton. If I do this way: $$\int { \frac { dx }{ x } } =x\left( \frac { 1 }{ x } \right) -\int { xd } \left( \frac { 1 }{ x } \right) =1+\int { \frac { dx }{ x } } ...
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0answers
13 views

Examples of physical motivation for integrals over scalar field?

I'm looking for good examples of physical motivation for integrals over scalar field. Here is an example I've found (source): A rescue team follows a path in a danger area where for each position ...
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0answers
27 views

How to relate the following function with Thomaes function

I am wondering how I can go about comparing the following functions $f(x)=\begin{cases} 1/n &\text{if $x=\frac{1}{n}$} \\ 0 &\text{else} \\ \end{cases}$ On the interval $[0,1]$, with the ...
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1answer
37 views

Splitting integral !!

i have this simple question that make me really confused : let $\phi$ a smooth function with compact support and $p>0,t\ge 0$ and : \begin{equation} U_p= \left\lbrace \begin{array}{ccc} 0 & ...
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1answer
26 views

parametric equations, finding the range of t

When parametrizing a curve how doe we obtain the range of $t$? For example lets say we have the parametrization: $x(t) = 1+3t$ and $y(t) = 2+5t$. How do we find the range of t? $t\to[?,?]$
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0answers
18 views

Definite integral of a continued fraction function

I came up with this function written as the following continued fraction (please correct me if my notation is incorrect): for $n\in\mathbb{N}$, let $$f(x;n)=x+\operatorname*{\LARGE ...
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0answers
20 views

Independence of parametrization in defining integral of differential form

This is an exercise from Spivak's Calculus on Manifolds. Questions asks the following : (Independence of parametrization). Let $c$ be a singular $k$ cube and $p:[0,1]^k\rightarrow [0,1]^k$ be ...
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0answers
18 views

Does the quadratic covariation process define a measure?

In the context of stochastic integration (when we define the space $L^2(M)$), we define the (possibly infinite) measure $$P_M = P \otimes [M]$$ by $$E_M[Y] = E\left[\int_0^\infty Y_s(\omega) ...
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0answers
45 views

Conjecture: $\int_0^{\infty}dx\frac{e^{i\alpha\sqrt{x^2+1}}}{\sqrt{x^2+1}}J_1(Qx)=\left(e^{i\alpha}-e^{i\sqrt{{\alpha}^2-Q^2}}\right)/Q$

Here $\alpha>0$, $Q>0$, and $J_1$ is a Bessel function. I'm fairly certain the closed form in the title is accurate for a couple of reasons. First, I've evaluated the integral numerically in ...
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2answers
36 views

ELI5: Riemann-integrable vs Lebesgue-integrable

I am wondering what the difference between riemann-integrable and lebesgue-integrable means. Does it have anything to do with the absolute value of the integrand, something like ...
1
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0answers
22 views

$E=\left\{(x_1,x_2,x_3,x_4):\sqrt{x_2^2+x_3^2+x_4^2}\le x_1\right\}$, what is $\int_E e^{-\langle x,t\rangle} \, dx$?

I'm learning some real analysis and encountered the following question: Let $E=\left\{(x_1,x_2,x_3,x_4)\in \mathbb{R}^4:\sqrt{x_2^2+x_3^2+x_4^2}\le x_1\right\}$. for which values of ...
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2answers
18 views

Rotational Volume

I have to find the volume of the region bounded by $ y= \sqrt{x-1} $, y=3, the y-axis and the x-axis rotated around y=5 I set up $\int_1^{10} $ $\pi((5-(\sqrt{x-1}))^2 - (5-3)^2)$dx + $\int_{0}^1$ ...
2
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0answers
48 views

The closed form of $\int^\infty_{B}e^{-(x+\frac{A}{x})}\,dx$, where $A>0$, $B>0$.

What tools, ways would you propose for getting the closed form of this integral? $$\int^\infty_{B}e^{-\left(x+\frac{A}{x}\right)}\,dx,$$ where $A>0$, $B>0$. When $B=0$, from Table of ...
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0answers
16 views

integration of product of error function

Is there a more simple formulae for the following integral $$ \int_{a}^{+\infty} erf(\alpha x).erf(\beta x) \frac{1}{x^2} \:\mathrm{d}x $$ where $a>0$, $\beta>0$ and $\alpha>0$
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1answer
44 views

Show $ (\int_{-\infty}^\infty \sqrt{p}\sqrt{q}d\mu)^2\leq 2 \int_{-\infty}^\infty \min\{p,q\}d\mu $

Consider a random variable $X$ in $(\Omega, \mathcal{F}, \mathbb{P})$. Let $p,q$ be two densities with respect to a measure $\mu$ in $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ where ...
4
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2answers
90 views

Difficult Integral $\int_0^{1/\sqrt{2}}\frac{\arcsin({x^2})}{\sqrt{1+x^2}(1+2x^2)}dx=$

I have a difficult integral to compute.I know the result, but need to know the method of calculation. How prove this result? ...
0
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2answers
43 views

How to get this result of integral?

Statement \begin{equation} \int_{\mathbb R} \exp \left( -2\pi (\frac{x}{\sqrt{2}})^2 \right) \exp\left( -i2 \pi \frac {x}{\sqrt{2}} \cdot f \right) dx = \exp \left( -\pi f^{2} / 2 \right) ...
2
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0answers
46 views

Sine improper integral

Suppose the following integral $$ \int\limits_{-\infty}^{\infty}\sin{x}dx $$ In mathematical rigor, the following is the definition $$ \int\limits_{-\infty}^{\infty}\sin{x}dx = ...
1
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4answers
54 views

Give that $f$ is a decreasing continuous function and that $f(x+y) = f(x) + f(y) -f(x)f(y)$ and $f'(0)=-1;$ Then find $\int_{0}^{1}f(x)dx$

Give that $f$ is a decreasing continuous function and that $$f(x+y) = f(x) + f(y) -f(x)f(y)$$ and $f'(0)=-1;$ Then it is to be found what is $\int_{0}^{1}f(x)dx$ I am at a loss on how to approach ...
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2answers
33 views

meaning of definite integral

So to my knowledge a definite integral's significance is how it shows the "intensity" or area under the curve for a function. However, I am confused then why the definite integral for x from 0 to 1 ...
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0answers
40 views

Prove that a function is Riemann integrable directly, using $\epsilon-P$

I know there already are questions like these, but I still don't understand how to prove it. Question: Prove that $f$ is Riemann integrable on $[0,1]$ if $$f(x) = \begin{cases} x^2 \sin (1/x) ...
2
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3answers
23 views

Substitution and Partial Fractions (Integration)

$$\int\frac{dx}{x-\sqrt[4]{x}}$$ given the substitution $x=u^{4}, dx=4u^{3}du$ $$=\int\frac{4u^{3}du}{u^{4}-u}=\int\frac{4u^{3}du}{u(u^{3}-1)}=\int\frac{4u^{2}du}{(u^{3}-1)}$$ At this point I ...
1
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1answer
47 views

How do you differentiate the integral from $ \int_{e^{-x}}^{e^x} \sqrt{1+t^2}\,dt$ [duplicate]

How do you differentiate the integral from $e^{-x}$ to $e^x$ of $\sqrt(1+t^2)$ with respect to t? $$ \int_{e^{-x}}^{e^x} \sqrt{1+t^2}\,dt $$ I know the answer is $$ e^x\sqrt{1+e^{2x}} + ...
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1answer
54 views

To refute : a function with one discontinuity point is integrable on $\left[0, 1\right]$

If, let $f: \left[ 0, 1\right] \to \mathbb{R}$ continuous with only one discontinuity point is integrable on $\left[ 0, 1 \right]$. I think this is false but I can't find a example that contradicts ...
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1answer
33 views

How to calculate the integral of exponential of complex exponential?

How to express $$\int^{\pi}_{-\pi}e^{c \cdot e^{j\omega}}d\omega$$ in closed form, where $c$ is a constant? Should it be some Bessel function?
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1answer
25 views

References for the “extended” Green and Stokes' theorem.

I was watching these videos from MIT's series: Green, Stokes. And I didn't understand the justification: their "extended version" of the theorems. I looked up on google and couldn't find many ...
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0answers
19 views

Riemann-Stieltjes Integral Substitution

I want to prove $\int^b_a\,f(g(x))\,dg(x) = \int^{g(b)}_{g(a)}\,f(x)\,dx$ for all f continuous. Firstly, $\int^b_a\,f(g(x))\,dg(x) = \int^b_a\,f(g(x))g'(x)\,dx$, since g is continuous and ...
1
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1answer
16 views

Error Bounds with Trapezoidal Formula

I know there are some posts about the same thing but I am unable to do my specific question or at least, I don't think I'm doing it the right way. The question asks me to use the Trapezoidal Error ...
3
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0answers
78 views

tough definite integral: $\int_0^\frac{\pi}{2}x\ln^2(\sin x)~dx$

Any ideas on $\int_0^\frac{\pi}{2}x\ln^2(\sin x)\ dx$ ? Best numerical approximation I can get is $0.2796245358$ Is there even a closed form solution?
0
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1answer
69 views

Closed form for this integral $I=\int_0^{1}\frac{{\arcsin}({x^2})}{\sqrt{1-x^2}}dx$

I’m trying to find a closed form for this integral.Any help is appreciated.Thanks $$I=\int_0^{1}\frac{{ \arcsin}({x^2})}{\sqrt{1-x^2}}dx$$
3
votes
1answer
56 views

A general case - Leibniz rule

Given the Leibniz rule: $$\frac{d}{dy}\int^a_b f(x,y) dx = \int_b^a \frac{\partial f}{\partial y} (x,y) dx$$ How do I prove a more general case using the chain rule and the above: $$\frac{d}{dy} ...
0
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0answers
24 views

Changing the order of integration? - $ \int_{-5}^{5}dx\int_{-7}^{\sqrt{25-x^2}}f(x,y)dy$

I'm trying to change the order of the integral and I don't understand it very well. I am trying to understand it by following solution to another example but still - I would appreciate any hints or ...
0
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0answers
24 views

what is $\int_{\gamma}\frac{2}{(z+2)^2}dz$ with $\gamma(t)=t+it\sin(\frac{\pi}{t})$ for $t>0$?

Again a question about integration. Consider the integral $$\int_{\gamma}\frac{2}{(z+2)^2}dz,$$where $\gamma:[0,1]\to\mathbb{R}$ such that $\gamma(0)=0$ and $\gamma(t)=t+it\sin(\frac{\pi}{t})$ if ...
2
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0answers
51 views

Inner Product of Chebyshev polynomials of the second kind with $x$ as weighting

I have tried to solve the integral $${\int_0^1 U_n (x) U_m (x)x dx },$$ where ${U_n (x) }$ denotes Chebyshev polynomial of the second kind. Solving the integration and checking the result, I ...
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1answer
20 views

Minimum of four exponential variables

Four accidents occur independently, with each accident following an exponential distribution with a mean of 22.5. What is the expected value of the minimum of the four accidents? Attempt: ...
0
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1answer
20 views

Double integral over the set with an absolute value of $y$

I need to calculate an integral over the set: $$D \colon 0\leq x\leq \pi\text{ and }|y|\leq x$$ from the set (definite integral) $D \int \cos(y)dA$ I don't understand what $|y| \leq x$ means. Can ...
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1answer
21 views

How can I calculate this integral of a differential form in a surface?

I'm trying to integrate the 2-form $\omega = A(y) dx \wedge dy - dx \wedge dz + B(z)dz \wedge dy $ in the set $R_f=\{(x,y,z),\quad z=f(x,y)\quad x^2+y^2 \neq 1 \}$ with $f$ a differentiable function ...
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0answers
24 views

Change of variables for path integral.

Let $G=C^\infty([0,1];\mathbb{R}^d)$ be smooth paths, then for the path $A\in G$, consider the translation operator from $G$ to itself $T_A:G\to G$ $$T_A(g)(t):=g(t)+A(t).$$ Does there exist a ...
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1answer
26 views

What happens to Chebyshev polynomials integration when n=1

The integration of Chebyshev polynomials of the first kind has the following value, $$\int T_{n}(x) \, dx = \frac{1}{2} \, \left( \frac{T_{n+1}(x)}{n+1} - \frac{T_{n-1}(x)}{n-1} \right)$$ what happens ...
1
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1answer
70 views

$\lim_{n \to \infty }\int_{0}^{n}\frac{n \cdot e^{\frac{x}{n}}}{x^4+n^2}dx=$?

$$\lim_{n \to \infty }\int_{0}^{n}\frac{n \cdot e^{\frac{x}{n}}}{x^4+n^2}dx=?$$ I am allowed to used all the classical techniques of calculus, and this was a question from measure theory when we were ...
0
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2answers
50 views

Solve $\int \frac{(\sin x)^2}{a+b\cos x}dx$ [on hold]

Can anyone help me to solve this? Thanks.
6
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2answers
74 views

$\lim_{n \to \infty} \int_{0}^{n}(1-\frac{3x}{n})^ne^{\frac{x}{2}}dx$=?

$$\lim_{n \to \infty} \int_{0}^{n}\left(1-\frac{3x}{n}\right)^ne^{\frac{x}{2}}dx$$ I thought about using the theorem of monotonic convergence and had ...
1
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1answer
20 views

Numerically integration with a an infinite upper limit and non-zero lower limit

I have seen lots of quadrature formulas where we have definite limits or one of the limits is infinity and the other is zero. But what about the following case $$f(x) = \int_a^\infty e^{\frac{x}{t}} ...
1
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1answer
29 views

Integration of $\frac{dx}{(6x-4x^2)^{1/2}}$ and completing square.

Integrate $$\frac{dx}{\sqrt{6x-4x^2}}$$ While completing the square of $6x-4x^2$, I want to know where did $9/16$ come from in the following after taking $4$ out as the common factor $$-4\bigg(x^2 ...
0
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0answers
20 views

Laplace transform of a definite integral

I'm having some troubles with what follows. I am interested in finding the Laplace transform w.r.t. $x$ of some real-valued, positive, continuous (in general well-behaved) function $f(x,t),x,t>0$. ...
0
votes
1answer
33 views

Do we have $\int_{[0,1]^3} \min (x,y,z)\,dx\,dy\,dz=6\int_0^1\int_0^y \int_0^z x \,dz\,dy\,dx?$

I have this nice question: $\int_{[0,1]^3} \min (x,y,z)\,dx\,dy\,dz .$ I think it equals $6\int_0^1\int_0^z \int_0^y x \,dx\,dy\,dz .$ Then using my formula I got the result to be $\frac {1}{4} $ ...