Tagged Questions

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Definite integral problem [on hold]

I've got a problem with integral: $$\int_{0}^{\pi/2}\sin((2n+1)x)\frac{\sin x-x}{\sin(x)\cdot x}\mathrm{d}x$$
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Why does the same limit work in one case but fail in another?

The following questions has been bugging me since high-school calculus. Please help me find my peace once and for all: Consider a revolution solid generated by rotating a nice curve $f(x)$ around the ...
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Riemann Integrable function, is this Riemann Integrable?

Define $f:[0,1]\rightarrow\mathbb{R}$ by setting $f(0)=0$ and $f(x)=\cos({1/x})$ when $x\not=0$ Use Riemanns Criterion to prove $f$ is integrable on $[0,1]$ I know that for a bounded function to be ...
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Geometric question involving integral of a function and its inverse.

I am given a function $\phi(s)$, continuous and strictly increasing with $\phi(0) = 0$, and want to show that for all $a,b \geq 0$, $$ab \leq \int_0^a \phi(x)dx + \int_0^b \phi^{-1}(x)dx.$$ I know how ...
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Determining function given it's domain, that it is uniform, and it evaluates to as a double integral?

I have a uniform probability distribution with density function $f(x,y)$ such that $$\int_0^2 \int_0^2 f(x,y)dy dx = 1$$ Now I know that $f(x,y)=\frac{1}{4}$ simply by considering the dimensions of ...
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Evaluating $\int_{0}^{1}\frac{\arcsin{\sqrt{x}}}{x^4-2x^3+2x^2-x+1}\operatorname d\!x$

Find this integral $$\operatorname I=\int\limits_{0}^{1}\dfrac{\arcsin{\sqrt{x}}}{x^4-2x^3+2x^2-x+1}\operatorname d\!x$$ My try: let $$f(x)=x^4-2x^3+2x^2-x+1$$ I found ...
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Superior limit of integrals of entire functions

Let $f$ be an entire function on $\mathbb{C}$. If $f$ is not constant, then I want to prove \limsup_{R\to\infty}\int_{\lvert z\rvert=R}\lvert f(z)\rvert\,\lvert dz\rvert=\infty. ...
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$\int \frac{e^x}{\sqrt{2e^x+e^{2x}}}dx$ integration

$\int \frac{e^x}{\sqrt{2e^x+e^{2x}}}dx$ I have no idea how to do it:( What is more I don't know how to start it... Any ideas? I will be pretty grateful. I tried some tricks but no one works.
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Calculate the value of $\int_0^\frac{\pi}{6} \frac{\cos x \operatorname d\!x}{\sqrt{\frac{1}{4}-\sin^2x}}$

$$\int_0^\frac{\pi}{6} \frac{\cos x \operatorname d\!x}{\sqrt{\frac{1}{4}-\sin^2x}}$$ so $$\lim_{\epsilon->\frac{\pi}{6}} \int^{\epsilon} _{0} \frac{\cos x}{\sqrt{\frac{1}{4} - \sin^2x }}$$ ...
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Integral $\int_{0}^{\infty}e^{-ax}\cos (bx)\operatorname d\!x$

I want to evaluate the following integral via complex analysis $$\int\limits_{x=0}^{x=\infty}e^{-ax}\cos (bx)\operatorname d\!x \ \ ,\ \ a >0$$ Which function/ contour should I consider ?
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Show that $dx = \frac{2}{1 + u^2} du$ where $u = {\tan(\frac{x}{2})}$

Hello everyone I have been trying to show that $dx = \frac{2}{1 + u^2} du$ where $u = {\tan(\frac{x}{2})}$ but I keep ending up with something like this: $2d{\sin(\frac{x}{2})}\cos(\frac{x}{2})$ ...
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Integral in $n-$dimensional euclidean space

I want to calculate this integral in $n$-dimensional euclidean space. $$I(x)=\int_{\mathbb{R}^n}\frac{d^n k}{(2\pi)^n}\frac{e^{i(k\cdot x)}}{k^2+a^2},$$ where $k^2=(k\cdot k)$, ...
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The sequence of indefinite integrals of a uniformly convergent sequence, converges uniformly

Question. Let $(g_n)$ be a sequence of Riemann-integrable functions, $$f_n(x)=\int_a^x {g_n(t)} \, \mathrm{d}t,$$ and $(g_n)$ uniformly convergent on $[a,b]$, then $(f_n)$ converges uniformly on ...
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$|f(z)|^{2}\leq \frac{1}{\pi r^{2}}\iint_{D(z,r)} |f(\theta)|^{2}dm(\theta)$ for $f \in H(\Omega)$

Let $\Omega$ be a domain ,$\overline{D(z,r)} \subset \Omega$, $f$ holomorphic in $\Omega$. a) Show that $$|f(z)|^{2}\leq \frac{1}{\pi r^{2}}\iint_{D(z,r)} |f(\theta)|^{2}dm(\theta)$$ where $dm$ ...
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Laplace transform and majorant

I am looking for a majorant such that for every $t>0$ we have that for all $x>0: |x^ne^{-xt}|\le F(x)$ such that $\int_0^\infty F(x) dx < \infty$? I guess this one does not exist, but the ...
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Improper parametric integral and differentiation under the integral sign

While looking at an astrophysic problem, I encountered the following integral $$\rho_{\infty} (r) = \int_{r}^{a} \frac{\rho_{0} (r_{0})}{\sqrt{r_{0}^{2} - r^{2}}} d r_{0} \;\;\;\;\;\;\; (1)$$ The ...
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Improper Riemann integral and imaginary exponential of real polynomials

Let $P(x_1,\cdots,x_n)$ be a real polynomial of degree $\geq 2$. What are the conditions on $P$ so that $$I_P:=\int_{\mathbb{R}^n} e^{iP(x)} dx$$ exists as an improper Riemann integral ? Already ...
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Integral $\int_{0}^{+\infty}\frac{t \sin(t)}{t^{2}+b^{2}}dt$

I want to solve the integral $$\int_{0}^{+\infty}\frac{t \sin(t)}{t^{2}+b^{2}}dt$$ Which function and contour should I consider ?
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Looking for an elementary solution of this limit

I was collecting some exercises for my students, and I found this one in a book: compute, if it exists, the limit $$\lim_{x \to +\infty} \int_x^{2x} \sin \left( \frac{1}{t} \right) \, dt.$$ It seems ...
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Function which is bounded and continuous except on a finite set of points is Riemann Integrable

I am trying to solve the following problem (Problem 7.2.15 of Bartle/Sherbert Book: Introduction to Real Analysis). The problem says: If $f$ is a bounded function on $[a,b]$, and there is a finite ...
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how to solve this integral in survival analysis

Let $T$ be a positive random variable, $S(t)=P(T\geq t)$. Prove that $$E[T]=\int^\infty_0 S(t)dt.$$ I have tried this unsuccessfully.
Calculate $\int_{\mathbb{R}} f(x)\delta(x^4+(\alpha-x)^4-(\alpha-\alpha_1)^4-\alpha_1^4)dx.$
i want to calculate $$\int_{\mathbb{R}} f(x)\delta(x^4+(\alpha-x)^4-(\alpha-\alpha_1)^4-\alpha_1^4)dx,$$ where $\alpha$ and $\alpha_1$ indenpent of $x$. can anyone give me some suggetions?
How to calculate $\int_{\mathbb{R}} f(x)\delta(x^2+\alpha^2) dx?$
If $~\alpha\in\mathbb{R},$ $~~\delta~~$ is the Dirac's delta function. Then how to calculate $$\int_{\mathbb{R}} f(x)\delta(x^2+\alpha^2) dx?$$