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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4
votes
3answers
111 views

Find all continuous functions $f:[0,1]\rightarrow \mathbb{R}$ that satisfy: $\int_0^1 f(x)dx=1/3 + \int_0^1 f^2(x^2)dx$

(Note that $f^2(x)=f(x)\cdot f(x)$ and not composition.) Since both integrals are defined, derivation is out of the question. I tried integrating the second integral by parts but reached something ...
3
votes
1answer
67 views

Why are functions that are continuous over $[a,b]$ integrable over $[a,b]$?

Why are functions that are continuous over $[a,b]$ integrable over $[a,b]$? Why is it that to be Riemann-integrable the infimum of the upper sums and the supremum of the lower sums have to be equal? ...
0
votes
0answers
23 views

Parametric functions to describe the intersection of two orthogonal cylinder surfaces?

I am trying to find a parametric equation for the intersection line of the surface of two orthogonal cylinders, $\vec{P}$ is a point that belongs to this intersection: $$\vec{P(t)} = \begin{bmatrix} ...
-1
votes
0answers
14 views

Is there any equality for the integral of the product of normal derivative?

I am trying to get the proof of $\int\int_DD_uf(x) D_ug(x) dx$. For example in Green Theorem, in integral we use the product of $ \nabla$, when it comes to normal derivative, how can I organize the ...
1
vote
0answers
27 views

Calculate the curvilinear integral

I need to calculate the curve integral. This should be the curve integral of I rank, which can be calculated with the formula : $$\int_{C}f(x,y)ds=\int_{a}^{b}f(g(t),h(t)) \sqrt{(\frac{dx}{dt})^2+\...
3
votes
1answer
96 views

A puzzle about integrability

I know there is a Proposition: for $f(x)$ is bounded on $[a,b]$,then $f(x)$ is integrable if and only if given $\epsilon>0$ ,there exists a partition such that $U(f,P)-L(f,P)<\epsilon$ But my ...
2
votes
2answers
116 views

Evaluation of Irrational Integral

Evaluation of $$\int\frac{x^4}{(1-x^4)^{\frac{3}{2}}}dx$$ $\bf{My\; Try::}$ Let $$I = \int\frac{x^4}{(1-x^4)^{\frac{3}{2}}}dx = -\frac{1}{4}\int x\cdot \frac{-4x^3}{(1-x^{4})^{\frac{3}{2}}}dx$$ ...
0
votes
4answers
105 views

Value of $\int\tan^{-1}(x)\,dx$

What is the value of $\int^{1000}_{0}\tan^{-1}(x)\,\mathrm d x$? Today we were taught about graphs of all trigonometric inverse functions. So my proofessor split it into $0-\tan(1)$ and $\tan(1)-...
0
votes
4answers
119 views

Strange integral result

Consider the following integral, $$\mathrm{I} = \int_{-1}^{1}\frac{d}{dx}\tan^{-1}\left(\frac{1}{x}\right)dx$$ We can do this in two ways, First Using the fact that the antiderivative of $\frac{d}{...
0
votes
0answers
11 views

Calculating the mean of the order statistics of Rayleigh random variables

I am trying to compute the mean of the ith order statistics for $n$ Rayleigh random variables as follows: $\int_0^\infty A \, x \, F^{i-1} (1-F)^{n-i} \, f \, dx$ where $A = i \binom{n}{i}$ and $F=1-...
1
vote
3answers
65 views

Interesting integral involving delta function.

How would one go about solving the following, $$\int_0^{\pi}\sin(x)\cdot\delta(\cos(x))\,\text{d}x.$$ I'm unsure where to begin. I am aware of the standard definition of the delta function in terms of ...
1
vote
0answers
33 views

Integration on manifolds with boundary

How can I define integral on manifolds with boundary? To use unity partition don't have I to deal with open sets of the same type, I mean, how can I be sure that there is a unity partition on my ...
4
votes
1answer
38 views

Is it possible to express an integral equation inside of a convolution

Given $$u(t) = \int_0^t y(\tau) d\tau$$ Consider a convolution type of integral $$W = \int_0^t\lambda^{t-\tau}y(\tau) d\tau$$ $\lambda$ a positive real number Is it possible to write $W = f(u(...
2
votes
1answer
72 views
+50

Drawing large rectangle under concave curve

Let $f$ be a continuous concave function on $[0,1]$ with $f(1)=0$ and $f(0)=1$. Does there exist a constant $k$ for which we can always draw a rectangle with area at least $k\cdot \int_0^1f(x)dx$, ...
1
vote
3answers
29 views

Evaluate integral over path using parametrisation

Evaluate the integral of $(F.dr)$ over the path (0,0,0) -> (1,1,1) where $F=e^{-x}i +e^{-y}j + e^{-z}k$ using parametrisation [x=t, y=t, z=t] I know from simpler questions in class that you must find ...
1
vote
3answers
68 views

Average value of $f(x)=\int_x^1 \cos(t^2) dt$ on the interval $[0,1]$.

Find the average value of the function $$f(x)=\int_x^1 \cos(t^2) dt $$ on the interval $[0,1]$.
1
vote
2answers
47 views

Compute antiderivative of $x \frac{3}{4} \sqrt{ -x}$

I need to compute the area under $x \times\frac{3}{4} \times \sqrt{|x|}$ from x=-1 to x=1. Therefore I need to compute the antiderivative of $x \times \frac{3}{4} \times \sqrt{x} $ for $0 \le x \le 1$,...
3
votes
2answers
66 views

Drawing large rectangle under curve

Let $f$ be a continuous nonincreasing function on $[0,1]$ with $f(1)=0$ and $\int_0^1 f(x)dx=1$. Does there exist a constant $k$ for which we can always draw a rectangle with area at least $k$, with ...
4
votes
3answers
123 views

Does $\int_0^2\frac{1}{\ln(x)}dx$ converge?

Given the following integral: $$ \int_0^2 \frac{1}{\ln(x)} dx $$ Does it converge? Iv'e gone this far: $$ \int_0^2 \frac{1}{\ln(x)} dx = \int_0^1 \frac{1}{\ln(x)} dx + \int_1^2 \frac{1}{\ln(x)} dx $...
-1
votes
0answers
28 views

Area of cross section in the intersection of two cylinders

A Steinmetz solid "B" is formed on the intersection of two cylinders, given as: x^2 + z^2 = r^2 and y^2 + z^2 = r^2 Although, referring to Gardner's work, I have successfully calculated the total ...
1
vote
1answer
90 views

A two-dimensional integral related to a Gaussian distribution

I am trying to evaluate the integral $I=\int_a^b\int_a^b\frac{1}{\sqrt{2\pi}\theta}e^{-\frac{(x-y)^2}{2\theta^2}}dxdy$. With the aid of Mathematica software, the result is $I=\left(e^{-\frac{(a-b)^...
1
vote
3answers
132 views

How to integrate $\int _1^{\infty }\frac{dx}{\left(x^2+1\right)\sqrt{x^2-1}}= \;?$

How do I integrate $\int _1^{\infty }\left(\frac{1}{\left(x^2+1\right)\sqrt{x^2-1}}\right)\:dx$? So what I've tried is substituting $x\:=\:\frac{1}{\sin t}$. So then I'll have that when $x\rightarrow ...
2
votes
3answers
92 views

Solve this indefinite integral: $\int \frac{x+2}{x^2-1}dx$

I need to solve this indefinite integral: $$\int \frac{x+2}{x^2-1}dx$$ but the result is wrong. Here my steps: $D° > N° $ so I need to reduce the Denominator, which is a difference of two square:...
3
votes
1answer
68 views

Statements about derivatives and integrals [on hold]

My professor gave me one example. It's given one intervall $I=\left [ a,b \right ]\subset \mathbb{R}$ and one function $f:I\mapsto \mathbb{R}$. There is also given 8 statements about derivatives ...
0
votes
2answers
28 views

Convergence of integral with absolute function

Given that the following integral converges: $$ \int_{0}^\infty |f(x)| dx$$ Prove or disprove that the following integral also converges: $$ \int_{0}^\infty f(x) dx$$ I thought to use the squeeze ...
0
votes
0answers
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 ( ...
0
votes
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 ...
1
vote
2answers
100 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 $ ...
0
votes
0answers
19 views

differentiation under the integral sign and change of variables

Let $f \in C^2 (\mathbb{R}^2)$ 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,\infty)}f_\phi(...
3
votes
1answer
119 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 ...
0
votes
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?
4
votes
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 ...
-1
votes
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(...
0
votes
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 ...
0
votes
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 ...
0
votes
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/\...
1
vote
3answers
133 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. ...
1
vote
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: $$f \ast g = \int_0^t \alpha e^{-\beta (t-\tau)} y(\tau) d\tau$$...
0
votes
0answers
62 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?
0
votes
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
votes
4answers
62 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!
1
vote
0answers
93 views

Prove that $I=J$

Motivation from China cat I particularly like the proof of Felix Marin(shot two birds in one stone) Show that $I=J$ then using only one of them to prove that it is equal to ${\pi\over 3\sqrt3}$ like ...
3
votes
4answers
126 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 ...
0
votes
1answer
62 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 ...
2
votes
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(...
2
votes
3answers
130 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
votes
4answers
114 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?
1
vote
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 $$\lim_{x \to \infty}{\frac{1}{x}\int^{3x}_{x/3}} g(t)\, dt?$$
-1
votes
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 ...