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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8 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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16 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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15 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
16 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
14 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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40 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
34 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 ...
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16 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
15 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 + $\int_{0}^1$ ...
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43 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
15 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 ...
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2answers
71 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? ...
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2answers
42 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) ...
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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 = ...
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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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22 views

Prove that a function Riemann integrable is [on hold]

I know there are questions like these, but I still don't understand how to prove it. Question 1 Do I use an epsilon proof or do I use the method of showing that $$\sup L(P, f) = \inf U(P,f)$$ ...
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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 ...
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1answer
45 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
49 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
30 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 ...
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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 ...
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76 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?
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1answer
58 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
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1answer
54 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} ...
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0answers
23 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 ...
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0answers
22 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 ...
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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
19 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: ...
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1answer
19 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
20 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
25 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 ...
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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 ...
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2answers
49 views

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

Can anyone help me to solve this? Thanks.
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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 ...
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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}} ...
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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 ...
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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 $f(x,t),x,t>0$. ...
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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} $ ...
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1answer
54 views

Is $\int_{|z|=2}\frac{z}{(z-3)^2}dz=0?$

I have a question. What is $$\int_{|z|=2}\frac{z}{(z-3)^2}dz?$$ In my optinion it must be zero, because the singularity $3$ is outside $\{z\in\mathbb{C}:|z|<2\}$, is it correct? Regards
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3answers
86 views

How to solve $\int \frac{1}{1-y^2}$ with respect to $y$?

I was solving an A Level paper when I came across this question. I tried substitution, but I'm not getting the answer with that. Would appreciate it if someone would help me.
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1answer
18 views

Evaluate [$-x^{n+1} e^{-x} ]_{x=0}^{x=\infty}$

Evaluate the following: [$-x^{n+1} e^{-x} ]_{x=0}^{x=\infty}$, where $n$ is any integer greater than 1 Any help?
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2answers
75 views

Prove $\int x^n\,dx=\frac{x^{n+1}}{n+1}.$ [on hold]

How to prove $\int x^n\,dx =\frac{x^{n+1}}{n+1}.$ I know integration is area under curve but how to continue any ideas.
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5answers
70 views

Explain solution to $\int\frac{ 7\,dx}{x(x^4 + 2)}$.

Can someone please explain how they get from the step outlined in red to the one in blue? I tried using partial fractions to break it up but that didn't work, and I'm not sure how else I could ...
0
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1answer
41 views

Convert Riemann sum to definite integral: $\sum_{i = 1}^n \frac{n}{n^2 + i^2}$

I am having trouble with this problem. Basically, I am given a Riemann sum and I have to rearrange it so that I can deduce the definite integral that it is equivalent to. Thank you. $$\lim_{n \to ...
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3answers
55 views

What are the lightest hypothesis needed to be able to get the limit inside the integral?

Let $\{f_n\}$ be a sequence of Riemann integrable functions. What are the lightest conditions on $f_n$ to guarantee the following? $$ \lim_n \int_a^b f_n\,\mathrm dx=\int_a^b \lim_n f_n \, \mathrm ...