Questions about the evaluation of specific definite integrals.

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${\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$ and $\int_{0}^{\pi/2} \frac{\log \cos x}{x^2}\:\mathrm{d}x$

I have found the following new result connecting to rational log-cosine integrals. Proposition. \begin{align} \displaystyle & {\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} ...
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Normalizing a probability density function

I need to find a normalization term $N(\alpha,\beta)$ for the probability density function: $$PDF(\alpha,\beta)=(x-x_1)^{\alpha}e^{-\beta(x-x_1)}$$ In other words, solve the following equation: ...
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Is there an analytical solution to the following Gaussian integral?

I wonder if there is an analytical solution to $$\int_{-\infty}^{\infty} \frac{e^{-x^2}}{(x+a)^2+b} dx,$$ where $a, b>0$. I know, of course, that the antiderivative of the fraction is a version ...
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Find $x > 0$ for which $\int_{0}^{x} [t]^2 \ dt = 2 (x-1)$.

What are all possible $x > 0$ for which the following equation is satisfied? $$\int_{0}^{x} [t]^2 \ dt = 2 (x-1),$$ where $[.]$ denotes the bracket (or floor) function. I guess we will have to ...
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How to solve this seemingly simple triple integral?

$$\iiint_D x^2+y^2+z^2\,dxdydz$$ $D$ is bound by $x=0, y=0,z=0$ and $x+y+z=a$, calculated by rote, I got $\frac{a^5}{20}$, is there any simpler way to do this? I tried using spherical coordinates, but ...
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Prove that $f$ is constant on $[a,b]$

$\displaystyle \int_{a}^{b} f^2(x) \, \mathrm{d}x$ = $\displaystyle \int_{a}^{b} f^4(x) \, \mathrm{d}x$ = $\displaystyle \int_{a}^{b} f^3(x) \, \mathrm{d}x$ And $f$ is continious on $[a,b]$ and ...
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The volume is preserved by the flow: where is the absolute value?

Consider the following excerpt of the Liouville's theorem proof taken from "Arnold - mathematical methods of classical mechanics": In changing the variables in the integral, I don't understand why ...
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Solution to the integral?

What is the solution of the following integral: $$ \int_{-1}^{K} x^{B+1} e^{-Nx} dx $$ where $N$ and $B$ are constants
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consider a square of side length $x$, find the area of the region which contains the points which are closer to its centre than the sides.

Any ideas how to start. I am having trouble figuring out the region itself All ideas are appreciated thanks
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Integral equation/ODE

I have to find all the functions $f(x)$ such that $$f(x)=xe^{(1-x^{2})/2}-xe^{-x^{2}/2}\int_{1}^{x}t^{-2}e^{t^{2}/2}dt$$ which satisfies $$f(x)=1-x\int_{1}^{x}f(t)dt$$ I tried to equal both, but when ...
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Find $\int_0^4\int_{0}^{4}xy \sqrt{1+x^2+y^2} \,dy\, dx $

I am having a tough time figuring this one out. Any help will be appreciated. do we have to approximate, or can we actually find it
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Volumes of Revolution Washer Method

I have to find the volume of revolution of a region called $C$ using around the $y=-1$ axis. The region is bounded above by $y \ = \ \ln(x+1)$, bounded below by $y=e^{-x}$ and on the right by $x=3$. ...
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if $ f(x)=x+\cos x $ then find $ \int_0^\pi (f^{-1}(x))\text{dx} $?

I would be interest to show : if $ f(x)=x+\cos x $ then find $ \int_0^\pi (f^{-1}(x))\text{dx} $ ? my second question that's make me a problem is that : what is :$ f^{-1}(\pi) $ ? I would be ...
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Fastest way to integrate $\int_0^1 x^{2}\sqrt{x+x^2} \hspace{2mm}dx $

This integral looks simple, but it appears that its not so. All Ideas are welcome, no Idea is bad, it may not work in this problem, but may be useful in some other case some other day ! :)
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What is the solution of the following integral? [duplicate]

I tried to solve the following integral using Maple as well as by hand but unable to do so. Can anybody help me in solving the following integral? $$ \int_{0}^{R} D\pi r^2 (D\pi r^2-1)^B 2\pi \lambda ...
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Approximate closed form of Integration involving a lot of terms

I tried to solve the following integral using Maple as well as by hand but unable to do so. Can anybody help me in solving the following integral? $$ \int_{0}^{R} D\pi r^2 (D\pi r^2-1)^B 2\pi \lambda ...
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symbolic definite integral using matlab

I have this function $$ \frac{di}{dt} = - \frac{R}{L} i $$ I know the solution which is $$ i(t) = i_{0} e^{- \frac{R}{L} (t-t_{0})} $$ I would like to get the same solution by using Matlab. How can ...
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closed form for $\int_0^1x^{a+1}(1-x^2)^bJ_a(cx)dx$

$$\int_0^1x^{a+1}(1-x^2)^bJ_a(cx)dx$$ my friend posted the integral on the fb and I tried to solve it but I faild because I have little information about bessel function so could some one help ?
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How to evaluate this integral?

How to evaluate $$ \int_{a}^{b} [x] \ dx \ \ + \int_{a}^{b} [-x] \ dx \ ? $$ I know that $[-x] = -[x]$ if $x$ is an integer, whereas $[-x] = -[x] - 1$ if $x$ is not an integer. So is it about ...
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Evaluating $\dfrac{1}{\Gamma (r)}\int_{0}^{x}(x-t)^{\alpha -1}t^{\lambda}dt$ [on hold]

How can I evaluate the following integral $$\frac1{\Gamma(r)}\int_0^x(x-t)^{\alpha-1}t^\lambda\ dt$$
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Choose appropriate contour for a complex integral

I have a problem to solve integral $$ I = \int^{\infty}_0 \frac{\mathrm{d}x}{(x-z)(1+x^2)^{\kappa+2}} $$ I can solve the same integral with borders $-\infty$ to $\infty$ using residue theorem but ...
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Arc Length in two dimensions by integration

I'm really at the end of my wits on this problem. Basically I'm trying to find arc length. The vector-valued function is: $R=\langle t,\sqrt{t}\rangle$ and $t\ge0$. We're looking for the length of ...
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Finding a root of a function via Rolle's theorem

Consider the function $f(t)=a(1-t)\cos(at)-\sin (at)$, where $a\in\mathbb R$. To show that it has a root in the unit interval I am urged to integrate $f$ and apply Rolle's Theorem. Attempt: $$\int ...
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Evaluation of $\int_0^1 \frac{\log(1+x)}{1+x}\log\left(\log\left(\frac{1}{x}\right)\right) \ dx$

I need some hints, clues for getting the closed form of $$\int_0^1 \frac{\log(1+x)}{1+x}\log\left(\log\left(\frac{1}{x}\right)\right) \ dx$$
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$\int_0^t m(x) \ dx =\int_0^t n(x) \ dx $ for all t > 0 implies that m(x) = n(x)

$$\int_0^t m(x) \ dx =\int_0^t n(x) \mathrm{d}x $$ For all $t > 0$. m and n are continuous. Prove that $m(x)=n(x)$ MY APPROACH I took f(x)= m(x)-n(x). Assume that f(x) is not zero at ...
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Generalized Fubini's theorem

Is there a generalized Fubini's theorem, which allows one to conclude that, for example, $$\iint_D \frac{dA}{x^2 - 2xy + 1} = \iint_{-1}^1 \frac{dx \, dy}{x^2 - 2xy + 1} = \iint_{-1}^1 \frac{dy \, ...
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Fourier transform of a sinusoidal function

Let us consider following table which I want to calculate myself $$ x(t)=\frac{\sin(\omega_bt)}{\pi t}\quad\iff\quad X(j\omega)= \begin{cases} 1 & \text{if $|\omega|<\omega_b$}, ...
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Compute $\int_{0}^{\infty}e^{-tz}(z+d)^{n-1}dz$ as a function of $\Gamma(n)$

Is it possible to compute this integral $$\int_{0}^{\infty}e^{-tz}(z+d)^{n-1}dz$$ as a function of complete gamma $\Gamma(n)$. If possible, I'm looking for a closed form solution. Thanks!
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Find limit $\lim\limits_{x \to \infty} \int_0^{x} \cos\left(\dfrac{\pi t^2}{2}\right)$

I looked at the graph and found that limit is $\dfrac{1}{2}$ And limit to $-\infty$ is $-\dfrac{1}{2}$ By the way, the function for which we are finding the limit is called Fresnel function
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Rewrite and approximate the sum as an Integral $\sum_{i=1}^{1000} \sqrt{i}$ [closed]

This is not an Infinite sum !, how do we change this to an Integral. $ $ We normally write an integral as an infinite sum.
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On an application of the Abel-Plana formula

Referring to a previous question, i am having a hard time trying to do the integral: $$-i\int_{0}^{\infty}\frac{\log \left[1+\frac{\left(s\log(1+ix) \right )^{2}}{4\pi ^{2}} \right ]-\log ...
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Double Integration: $\iint_D\ e^{30x}\ dA$

I am having trouble with this double integral. I know I must set it up to have the $y$ values go from $x$ to $x+1$ and the $x$ values from $0$ to $1$. When I solved the integral I got the answer ...
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Finding volume between plane and paraboloid

Find the volume between bounded by $z=4$ and $z=x^2+y^2$.(Answer: $8\pi$) I thouhg using dievergence theorm ($\iint_KdivFdxdydz=\iint_SF\cdot\hat{n}dS$) for $\vec{F}=\big(\frac x 2,\frac y ...
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Integrating $\iint_R \sin(9x^2+4y^2)\ dA$

The question I'm trying to solve is: $\displaystyle \iint_R \sin(9x^2+4y^2)dA$, where $R$ is the region in the first quadrant bounded by $9x^2+4y^2=1$. I'm a little confused in solving this. Does ...
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Evaluate $\int_0^\infty x^{\lambda-1} \exp\left(-ax-b\sqrt x-\frac{c}{\sqrt x} - \frac{d}{x}\right) \: dx$

Is there a closed form for the integral $$\int_0^\infty x^{\lambda-1} \exp\left(-ax-b\sqrt x-\frac{c}{\sqrt x} - \frac{d}{x}\right) \: dx?$$ where $\lambda>0$, $a>0$, $d>0$ and where $b$, ...
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Conditions for changing the order of integration for contour integral.

I assume an integral $$I=\int_0^\infty f(x)g(x)\mathrm dx \tag{1}$$ where the function $f(x)$ can be represented as a contour integral in complex plane: $$f(x)=\oint_\Delta ...
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I need help with this integral

Someone can help me with this integral please? $$ \int ne^{-n^2} dn $$ I tried with parts $$ u = e^{-n^2} \ \ \ \ \ \ \ \ \ \ v = n \ dn $$ $$ du = -e^{-n^2}2n \ \ \ \ \ \ \ \ \ \ \ v = ...
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Varying definition of Controllability Gramian

I have always used the following definition of controllability gramian of $(A,B)$ over the time window $[t_0,t_f]$ $$W_c[t_0,t_f] = \int_{t_0}^{t_f}e^{A(t_f - t)}BB'e^{A'(t_f - t)}$$ I have used ' as ...
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Double integral - calculating without translating a circle to $(0,0)$

So I'm a bit confused with calculating a double integral when a circle isn't centered on $(0,0)$. For example: Calculating $\iint(x+4y)\,dx\,dy$ of the area $D: x^2-6x+y^2-4y\le12$. So I kind of ...
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Volumes of the solid of revolution around y axis

The body which regards to the functions $\left\{0\le y\le x^{-2}\sin\left(\frac{1}{x}\right),\:x\in \left(0,1\right)\right\}$ is given. Please calculate the volume of the revolution of it around the y ...
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Convolution Integral

Does someone know how to solve this convolution integral? $$V(x)=c_1\int\limits_{-\infty}^\infty \left(r(\tilde x)+\dfrac{c_2}{n(\tilde x)}-n(\tilde x)\right)\left(\sqrt{c_3+(x-\tilde x)^2}-|x-\tilde ...
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Evaluating integral involving Bessel function.

Evaluate $$\int_0^{\infty } \frac{2^{\frac{r}{\delta }} \left(2^{\frac{r}{\delta }}-1\right) r\ e^{-\frac{\alpha ^2+\left(2^{\frac{r}{\delta }}-1\right)^2}{2 \beta ^2}}\log 2 }{\beta ^2 \delta } ...
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Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,\frac{t}{d})$, where $F$ is the Gauss' hypergeometric function

What is the Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,\frac{t}{d})$, where $\gamma >0 $ and $F$ is the Gauss' hypergeometric function. Note that I have the Laplace transform of : ...
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Prove that $s_n \leq 1+\ln n$, where $s_n$ is the $n$th partial sum of the harmonic series

This is a very Interesting question, there are many ways to do it. Lets see what is the best way to do it. I have an idea which involves a definite integral, I am working on it, will post it soon.
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Prove that $\lim\limits_{n \to \infty}\frac{x^n}{n!}=0$ [duplicate]

Well I can Intuitively see that. I am wondering If there is a neat way to prove that
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If $h$ is closer to $f$ than to $g$, its integral on $\{f > g\}$ must “agree” with $f$'s?

I have the following question (a result I would like to prove, with an admirable record of failing at it so far) -- I actually do not know if it is obvious or just plain wrong. Let $f,g,h\colon ...
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Does it hold? $\int_{0}^{\pi/2}\cos^{2k}xdx=\frac{(2k-1)!!}{2k!!}\frac{\pi}{2}$

Does it hold? $$\int_{0}^{\Large\frac\pi2}\cos^{2k}x\ dx=\frac{(2k-1)!!}{2k!!}\cdot\frac{\pi}{2}$$ where $k \in \mathbb Z$ I have seen some examples like $$\int_{0}^{\Large\frac\pi2}\cos^{6}x\ ...
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Difficult Surface Integral

I am trying to perform a surface integral over kind of a weird shape. So the radius of the shape should be equal to the multiple of $3$ constants (one for each of the $x, y$ and $z$ directions) each ...