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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42 views

Jensen's inequality for two random variable

Prove: Let $X$ and $Y$ be two random variables in probability space $\left ( \Omega ,\mathcal{F},\mathbb{P} \right )$ , and $f:\mathbb{R}^2\rightarrow \mathbb{R}$ is a convex function, then $$f\left ( ...
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1answer
35 views

Changing to Polar Coordinates for Area between $2$ Tangent Circles

I would like to calculate a double integral over: $\{(x,y) | 4x \leq x^2+y^2\leq 5x\}$. I am trying to change to polar coordinates. So the theta would go from $0$ to $2\pi$. But I am not sure what ...
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2answers
102 views

Closed form for $\int_0^1 d u \, \frac{1}{u + \lambda} \ln \left(\frac{1 + u}{1 - u} \right)$

The parameter $\lambda$ is complex and it's not on the real axis. There are some similar cases: Help me evaluate $\int_0^1 \frac{\log(x+1)}{1+x^2} dx$ Evaluate $\int_0^1 \frac{\ln(1+bx)}{1+x} dx $ ...
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0answers
26 views

Differentiation under the integral sign and change of variables

Let $f \in C^2 \left(\mathbb{R}^2\right)$ with a bounded support, and let $f_\phi (x,y)=f(x\cos{\phi}-y\sin{\phi},x\cos{\phi}+y\sin{\phi}))$. Show that: $$\frac{d}{d\phi}\iint_{\mathbb{R}\times(0,\...
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1answer
120 views

If $g$ is Riemann-integrable in a closed interval and $f$ is a increasing function in a closed interval, is $g\circ f$ Riemann-integrable?

If $g$ is Riemann-integrable in a closed interval and $f$ is a increasing function in a closed interval, is $g\circ f$ Riemann-integrable? To clarify: the problem stated that the composition is well ...
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1answer
33 views

Definite integral - Integration by parts [closed]

Let $p,f,g,q$ be continuous functions on $[a,b]$. How can I show that $$\int_a^b (pf'g'+qfg)dt=\int_a^b f(-(pg')'+qg)dt$$ Maybe by integrations by parts?
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2answers
170 views

How can I calculate $ \lim_{h\to 0} \frac{1}{h}\int_{2}^{2+h} F(x)\,dx$?

Let, say, $F(x) = \sin(x^2)$ which is continuous, therefore there exists a $c \in [2,2+h]$ such that $$ F(c) = \frac{1}{h}\int_{2}^{2+h} F(x)\,dx.$$ I'm trying to calculate the limit when $h$ goes ...
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0answers
26 views

square integrable function?

If I want to find out for which $\alpha$ the function $f:B_1(0)\to\mathbb{R}$, $f(x)=x|x|^\alpha$ is in $L^2(B_1(0))$, where $B_1(0)\subseteq \mathbb{R}^n$, can I do something like this: $$\int_{B_1(...
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4answers
65 views

Using Double Integral Find the volume of sphere $x^2 + y^2 + z^2= 4 $ cut by cylinder $\ x^2+y^2=2y $

Using Double Integral Find the volume of sphere $x^2 + y^2 + z^2= 4 $ cut by cylinder $\ x^2+y^2=2y $ , When i try to make integral the limits are: $\ -1<= x<=1 $ and $\ 0<=y<=2 $ ,but i ...
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2answers
38 views

How to antidifferentiate a function not applicable to basic antidifferentiation rules? [closed]

For example, how would one go about finding $\int (\pi(x)) dx $? Is there a certain technique or formula? That is, how does one antidifferentiate a function without using integration rules? How does ...
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1answer
31 views

Find the derivative of $x(t) = \int_0^t \lambda^{t-\tau} y(\tau) d\tau$ in one step

Given $$x(t) = \int_0^t \lambda^{t-\tau} y(\tau) d\tau$$ where $\lambda \in \mathbb{R}_{>0}$ Find $\dot x(t)$ Claim: The answer can be obtained in one step yielding $\dot x = y - \log(1/\...
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3answers
134 views

Evaluate $\int_{-\infty}^{\infty} \frac{(1-\cos { y } )}{\mid{y}\mid^{1+\alpha}}dy$ [on hold]

How do I evaluate the following integral? $$\int_{-\infty}^{\infty} \frac{(1-\cos { y } )}{\mid{y}\mid^{1+\alpha}}dy=\frac{\pi}{\Gamma(1+\alpha)\sin(\frac{\pi\alpha}{2})}$$ Thank you in advance. ...
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3answers
47 views

How to derive through a convolution?

Let $f(t) = \alpha e^{-\beta t}$, where $\alpha, \beta$ are constants Let $g(t) = y(t)$ Then the resulting convolution $f\ast g$ is: $$f \ast g = \int_0^t \alpha e^{-\beta (t-\tau)} y(\tau) d\tau$$...
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0answers
63 views

What is $\int_{-1}^0\sin(t)e^{t^4}\mathop{\mathrm{d}t}$? [closed]

$\int_{-1}^0\sin(t)e^{t^4}\mathop{\mathrm{d}t}$ Is there a way of solving this definite integral using simple methods?
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3answers
110 views

Integration by part for solving $\int \tan^{-1}\sqrt{x}\; dx$

I want to solve the intgral is given by $$\int_{0}^{1}\tan^{-1}\sqrt{x} \;dx$$ I set $dx=dv$ and $\tan^{-1}\sqrt{x}=u$ but I do not recieve good result. please give me hint
4
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4answers
66 views

How do I find the Integral of $\sqrt{r^2-x^2}$?

How can I find the integral of the following function using polar coordinates ? $$f(x)=\sqrt{r^2-x^2}$$ Thanks!
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4answers
127 views

Can $\int_0^{2\pi} \frac{dx}{\sin^6x+\cos^6x}$ be solved using $\cot x = u$ as substitution?

My first guess is it can't, since when I substitute the boundaries, I end up with $\cot2\pi$ and $\cot0$. Nevertheless I tried substituting pretending it is indefinite integral, but I can't get ...
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1answer
64 views

A Property of Integrals defined for step functions. [duplicate]

In Apostol's Calculus Volume-1 the proof of Additive Property for Integrals of Step Functions is given as an exercise that is: $$\int_a^b[u(x)+g(x)]dx=\int_a^b u(x)dx+\int_a^b g(x)dx$$ And Integrals ...
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2answers
38 views

for $f\in C^2(\mathbb{R})$, finding the derivative of $\frac{d}{dt}\int_0^\infty f(x+t)\cdot xdx$

Let $f\in C^2(\mathbb{R})$, (a) Prove that $$\frac{\mathrm{d}}{\mathrm{d}t}\int_0^\infty f(x+t)\cdot x\mathbb{d}x=-\int_0^\infty f(x)\mathrm{d}x$$ (b) Prove that $$ \iint_{(0,\infty)\times(...
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3answers
131 views

How to calculate $\lim_{n \to \infty} \int^{2007}_{0}e^{\frac{x^{2008}}{n}}dx$?

How to calculate $$\lim_{n \to \infty} \int^{2007}_{0}e^{\frac{x^{2008}}{n}}dx?$$ Can I just write $e^{\frac{x^{2008}}{n}} \rightarrow e^0$ when $n \to \infty$?
3
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4answers
117 views

If $\int_2^\infty f(x)^2 dx $ is convergent, is it true that $\int_2^\infty f(x)x^{-3/4} dx $ is convergent?

If $\int_2^\infty f(x)^2 dx $ is convergent, is it true that $\int_2^\infty f(x)x^{-3/4} dx $ is convergent?
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1answer
45 views

How to calculate $\lim_{x \to \infty}{\frac{1}{x}\int^{3x}_{x/3}} g(t) dt$?

Function $g: (0; +\infty) \rightarrow \mathbb{R}$ is unbounded, continous and has limit in $+\infty$ equal to $\pi$. How to calculate $$\lim_{x \to \infty}{\frac{1}{x}\int^{3x}_{x/3}} g(t)\, dt?$$
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2answers
50 views

Find the antiderivative…

Find the complete solution of the given differential equation $${dy \over dx} = {3x \sqrt{1+y^2} \over y}$$ I know how to solve it if the right side didn't contain either $x$ or $y$, but I can't ...
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2answers
39 views

Differential equation with one derivative: $y'=y\cos(x)+x\cos(x)-1$

$y'=y\cos(x)+x\cos(x)-1$, I tried to make it in the form $ay''+by'+c=0$, but I can't find the roots.
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2answers
101 views

Need help solving $\int\frac{\sin(x+1)\cos(x)}{\sin^2(x)+4} dx$

As the title says I need help solving the indefinite integral $$\int\frac{\sin(x+1)\cos(x)}{\sin^2(x)+4}dx$$ Thank you for any help.
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3answers
82 views

How to evaluate this integral $\int_{-a}^{a}{e^{-\iota k x}}dx$? [on hold]

I am having trouble to evaluate this integral. $$\int_{-a}^{a}{e^{-i k x}}dx=2\frac{\sin\left(a k\right)}{k}$$ Here $\iota$ stands for imaginary part. Any input will be appreciated.
2
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1answer
35 views

Differentiation under the integral sign, where the partial derivative of the integrand is not bounded by a Lebesgue integrable function.

Let $K(t)=\int_1^\infty u(t,x)\ \mathrm{d}x$, where $$u(t,x)=\frac{\cos{tx}}{x^2}\mathbb{1}_{[1,\infty)}(x).$$ I need to show that, for $t>0$, $$\frac{dK}{dt}(t)=\frac{1}{t}\left(K(t)-\cos{t}\right)...
2
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1answer
33 views

Show that $tE(1/X;X>t)\to0$ when $t\to0+$

Let $X\geqslant 0$ be a random variable,then $$\lim_{t\rightarrow 0+}{ \,\,t\int_{\left [ X> t \right ]} \frac{1}{X} \, {\mathrm{d}\mathbb{P}} }=0$$ I have no idea of how to prove it.
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0answers
44 views

Integral evaluation - Gamma distribution

I have a sequence of independent random variables which are $\chi^2(1)$ distributed, $(X_i)_{i=1}^n$, $X_i\sim\chi^2(1)$. If I consider the sum $\frac{t}{n}\sum_{i=1}^n{X_i}$ this should be $\sim\text{...
2
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1answer
103 views

$f(x)>0, f''(x)>0, 0<x<1, \int_0^1f(x)dx=1$, Prove: $\int_0^1|f(x)-t|dx\leqslant \frac{(1-t)^2+1}{2}, \forall t\in\mathbb R$

$\displaystyle f(x)>0, f''(x)>0, 0<x<1, \int_0^1f(x)dx=1$, Prove: $\displaystyle \int_0^1|f(x)-t|dx\leqslant \frac{(1-t)^2+1}{2},\quad \forall t\in\mathbb R$. I have tried this: $$\int |...
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3answers
48 views

How to find the integral that generates X number in Maple 17?

By a challenge of my calculus' professor, we have to find a integral that generates a 8 digit number, such as 12345678. Since there is a infinite number of integrals that could result in a 8 digit ...
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2answers
30 views

How to pull out coefficient from radical in an integral

I am in an online Calculus 2 class, and before my professor gets back to me, I was wondering if you guys could help. I was reading through an example: How was 1/27 pulled out from the coefficient ...
3
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1answer
17 views

Double integral of branch function when integrating with respect to the condition

In my probability class I have resorted to calculating a seemingly difficult integral for finding the expectation of a conditional expectation (law of total expectation) the function is: $$ F(\alpha, ...
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2answers
42 views

Calculating an easy iterated integral

I'm trying to solve the following integral: $$ \int_{0}^{t}dt_{1}^{'} \int_{0}^{t_1}dt_{2}^{'} \; \sin \; [k(t_1^{'} - t_2^{'})]. $$ The correct answer is: $$ \frac{\sin(kt) - kt}{k^2}. ...
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4answers
150 views

Different ways of evaluating $\int_{0}^{\pi/2}\frac{\cos(x)}{\sin(x)+\cos(x)}dx$

My friend showed me this integral (and a neat way he evaluated it) and I am interested in seeing a few ways of evaluating it, especially if they are "often" used tricks. I can't quite recall his way,...
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1answer
46 views

Differentiation under integral sign- Multivariable case problem

Let $f_{\theta}(x,y)=f(x\cos \theta-y\sin \theta,x\sin\theta+y\cos\theta)$, where $f\in C^2(\Bbb{R}^2)$(Is the range necessarily $\Bbb{R}^2$? This is quite ambiguous.) a function with a bounded ...
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1answer
74 views

How to find the Riemann sum for $\int_{\pi/8}^{\pi/4} \tan(x) \,dx$? [closed]

How can I evaluate the Riemann sum for $\int_{\pi/8}^{\pi/4} \tan(x) \,dx$ ? I don't know how to solve it without solving the integral in a standard way. Evaluate: $$\lim_{n\to\infty}\sum_{k=1}^{n} \...
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1answer
53 views

How we can solve like this integral $\int \frac{dx}{x^n+1}$ (Hypergeometric function) [duplicate]

In the wolfram given equation $\displaystyle\int \dfrac{dx}{x^n+1}=x\ {}_2F_1\left(1,\dfrac{1}{n};1+\dfrac{1}{n};-x^n\right)$, But what it's mean? How we can solve without this.
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38 views

How to solve: $\int\limits_{-\pi \cdot 0.5}^{\pi \cdot 0.5} \sin(x) \int\limits_{0}^{\cos(x)} e^{\sin(t)} \, dt \, dx$ [duplicate]

Let $ f(t) = e^{\sin(t)} $ and $ F(x) = \int\limits_0^x f(t) \, dt $ and then one has to find: $\int\limits_{-0.5 \pi}^{0.5 \pi} \sin(x) F(\cos(x)) \, dx $ Thus far I got: $$ I :=\int_{-0.5 \pi}^{...
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2answers
125 views

$ 2.28319 = 0 $ !? - How do I spot the mistake?

I am given $ f(t) = e^{\sin(t)} $ and $ F(x) = \int\limits_{0}^{x} f(t) dt $ and have to compute: $\int\limits_{-0.5 \pi}^{0.5 \pi} \sin(x) F(\cos(x)) dx $ How do I go about this problem? Thus ...
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0answers
40 views

Integrals of the form $\int x^m(a+bx^n)^Pdx$

I was reading a book on Integral Calculus, and in one chapter, the author dealt with methods of solving Integrals of the form $$\int x^m(a+bx^n)^Pdx$$ The author broke it down into 4 cases:$$$$ $...
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1answer
95 views

Evaluate improper integral: Exponent of square root.

I was working in a problem in physics, which gave me this integral, and I need solution: $$ \int_{-\infty}^\infty \exp{\left(-\sqrt{x^2 + a^2}\right)}dx $$ The problem is, I have no clue how to start....
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1answer
26 views

triple integral square pyramid

The pyramid has the base on the xy plane; vertices $(\pm 1,0,0), (0,\pm 1,0),(0,0,1)$ So basically, with my integration limits I thought I was calculating the volume of $\frac14$ of the pyramid when ...
2
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1answer
60 views

evaluate if integral converge & determine antiderivative

The problem is i need to study the convergence of A and B and find the antiderivative of C $$A=\int_0^\infty \frac{\sin(x) +x}{\sqrt x + x^3}dx$$ $$B=\int_0^\infty \frac{1}{\sqrt {e^x-1}(x^2+x^{1/...
2
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1answer
34 views

Triple integral in spherical / cylindrical coordinates - where's the error? Exercise check

I have done an exercise in two different ways but I have obtained two different results and I can't understand what's wrong. Please, help me. Given: $V=\{(x,y,z)\in R^3: x^2+y^2+z^2\leq 1, \frac{...
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0answers
33 views

Finding the maximum of $|\widehat{f''}|$ for $f$ in terms of the Gaussian

Let $$f = \begin{cases} e^{-x^2/2} - e^{-2 x^2} &\text{if $x\geq 0$,}\\ 0 &\text{if $x<0$.}\end{cases}$$ I would like to find out $|\widehat{f''}|_\infty$. A good numerical bound -- of ...
2
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2answers
112 views

I want to know if the way I derived the surface area of a sphere by integration is correct?

I am using the alias of Sillysack Buttowski and this is my first question. I searched on other links on stack exchange regarding "how to find the surface area of a sphere by integration". They seemed ...
3
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0answers
61 views

Integral over Julia Set (Is my math correct?)

So I was answering this question about whether or not the Julia Set was self-similar in a known way. Of course it is, and that got me thinking. Even though the self similarity is nonlinear, what if ...