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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5 views

$X= \int_0^S e^{(S-s)A} B u(s) ds \Rightarrow X= \int_0^T e^{(T-s)A} B \bar{u}(s) ds$

Consider the ODE system $$X'(t) = AX(t)+Bu(t)$$ where $X(t) \in R^n, \; A \in R^{n \times n} \text{ and } B \in R^{n \times m}$. In control theory, we define the set of states reachable as $$A(0,T) = ...
5
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2answers
29 views

Integral of $\sin(\pi x) \cos(n\pi x)$ and $\sin(\pi x) \sin(n\pi x)$

For an assignment, I am required to calculate the fourier series of this function: $$ \\ f(x) = \begin{cases} 0 & -1 < x < 0\\ \sin(\pi x) & 0 < x < 1 \end{cases} $$ To do this, ...
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1answer
20 views

Exponential Ingratiation solving

In the below definite integral, I am confused with the result, how they get 1/a for the integral part? Can we do in this way? $$\int_\infty^{-\infty} e^{-2a|x|}dx = \frac{1}{-2a} [e^{-\infty ...
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1answer
18 views

Proof by basic principles of Riemann Integration

Let $f:[a,b] \rightarrow \mathbb R$ be Riemann Integrable on $[a,b]$ and $f(x)\ge0$ for all $x\in[a,b]$. Show that $$\int_a^b f(x)dx\ge0$$ using basic principles of Riemann Integration. I'm new to ...
2
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3answers
55 views

Calculation of improper integral: $\int_{0}^{\infty}\frac{x^3\ln x}{(x^4+1)^3} \,dx$

One of an exam's task was to calculate the following integral: $$I=\int_{0}^{\infty}\frac{x^3\ln x}{(x^4+1)^3} \,dx$$ I tried integration by parts: $$I=\frac{1}{4}\int_{0}^{\infty}\ln x \cdot ...
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0answers
11 views

How to derive the compute complicated integrals given in integral tables?

I would like to evaluate or simplify a couple of integrals in the following manner $$\int_{0}^{\infty}x\sinh(\pi x) \cosh(x\cdot \beta)\Gamma\left( \frac{1}{2} +s +ix \right)\Gamma\left( \frac{1}{2} ...
1
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1answer
25 views

Area between $y = x^3 - 3x^2$, the $x$-axis and the lines $x = 2$, $x = 4$.

I was solving a problem today, and it appears my approach is at serious odds with the provided solution. The Problem Find the area between $y = x^3 - 3x^2$, the $x$-axis and the lines $x = 2$, $x = ...
0
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1answer
39 views

Integrating Floor Functions

I am having difficulties understanding and solving the following problem. So far I think I understand the first line. I know what the graph of floor[t] looks like and I understand they're asking to ...
1
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0answers
10 views

Adaptive Finite Element; Laplace equation

I'm currently attempting to turn my code for solving the laplace equation using finite element approximations into an adaptive one using the dual weighted residual as my error estimator: i.e. my ...
2
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2answers
21 views

Do roots lead to two antiderivatives that differ in their non-constant terms?

Consider the following example: $f'(x)= x^{-3/2}$ and $f(4)=2 $ $f'(x)= x^{-3/2}\Rightarrow \frac {x^{(-3/2) + 1}} {-1/2} \Rightarrow$ $\frac{-2}{\sqrt x} +C =f(x)$ This is where the problem ...
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0answers
40 views

Some questions about computing integrals?

I am trying to compute some integrals in the paper. My question is about the example on page 119 after Lemma 5.2.8 on page 118. Why $$\int_0^T e^{-\alpha_2(\pi)} = \int_0^T ds ...
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0answers
19 views

Definite integral of recursive functions

Let $F_0(x)=x, F_1(x)=4x(1-x), F_{n+1}(x)=F_1(F_n(x))$. Prove that for $n\in\mathbb{Z_0}^+$: $\int^1_0 F_n(x)\,dx = \frac{2^{2n-1}}{2^{2n}-1}$ Here is progress that I have: I used the substitution ...
2
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4answers
46 views

Integrate $\tan^2(\frac{\pi}{12} \cdot y)$

Integrate $\tan^2(\frac{\pi}{12} \cdot y)$ Wolfram gives the answer: $$\frac{12 \tan(\frac{\pi y}{12})}{\pi} -y + \text{constant}$$ But why is the value of $\tan^2$ not getting differentiated? ...
0
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2answers
33 views

Split the integral?

Let $ A = \int f(x)g(x)dx $, where $A$ is a real number, and let $A$ and $f(x)$ be known. Okay, so is there such a thing as 'splitting the integral'? I guess the broader question would be, is there ...
4
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1answer
69 views

How to integrate following indefinite integal?

The integral is $$ \int\frac{x-\sin x}{1-\cos x} \,dx $$ However, the only guess I have is that the denominator is the derivative of the numerator. Probably the integration by substitution will ...
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0answers
27 views

Conversion of proper to improper integrals and vice versa, is the following case possible?

I recently stumbled across a particular integral which got me asking myself this: Given a function (Smooth enough that we can play around with, and no convergence issues when integrating) $f(x)$, is ...
1
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1answer
28 views

Integral of cosine over a triangle

I need to integrate $\pi \cos (\pi x)$ over a triangle T with vertices $(0,0)$, $(1,1)$ and $(2,0)$. Me reasoning is: $$\pi \int_T \cos (\pi x) dxdy =\pi\int_0^1 \int_0^{y=x}\cos (\pi x) ...
1
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2answers
37 views

Evaluating a basic integral of the exponent.

Upon reading some mathematical literature, I have encountered the following computation: $x\in X$, a Banach space, $\alpha=\text{Re }(z)$ for $z\in\mathbb{C}$ and $\omega$ is the growth bound. ...
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2answers
47 views

Evaluating $\int_{\pi/4}^ { \pi/2} \frac{1}{\sqrt{1+\cos²(x)}} \text{d}x$?

I want to evaluate this integral $$\int_{\pi/4}^ { \pi/2} \frac{1}{\sqrt{1+\cos^2(x)}} \text{d}x $$ but I don't know how to proceed.
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2answers
116 views

Calculating in closed form $\int_0^{\pi/2} \arctan\left(\sin ^3(x)\right) \, dx \ ?$

It's not hard to see that for powers like $1,2$, we have a nice closed form. What can be said about the cubic version, that is $$\int_0^{\pi/2} \arctan\left(\sin ^3(x)\right) \, dx \ ?$$ What are ...
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0answers
11 views

To show that $L(f) = \sup \{L(P;f) : P \in P^*\}$.

Let $a>0$ and let $J = [-a , a]$. Let $f: J \to \Bbb R$ be bounded and $P^*$ be the set of all partitions $P$ of J that contain $0$ and are symmetric. Show that $L(f) = \sup \{L(P;f) : P \in ...
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1answer
35 views

Integral of the product of two functions, one of which is unknown. [on hold]

Okay: say I have two functions, and I know what one of them is, and I know what the value of the integral of the product of them is. Then: can I solve for the unknown function? I anxiously await ...
1
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2answers
33 views

I don´t get this result, of Trig Substitution in functions.

I got this integral: $\int\frac{2}{x\sqrt{x^2-5}}dx$ And, at the end I got: $\frac{2}{\sqrt{5}}arcsec\frac{x}{\sqrt{5}}+c$ Why the text's answer is? ...
0
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1answer
29 views

Calculate the flux through a surface S and my approach using Divergence theorem

Since my previous, introductory question Calculate the flux through a surface S from a field described by vectors about this example raised even more questions that I had initially - I was advised to ...
0
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3answers
84 views

Finding the antiderivative of $f(x)=\sqrt[3]{x^2+2x}$

$$f(x)=\sqrt[3]{x^2+2x}$$. Let $g(x)$ be an antiderivative of $f(x)$. If $g(5)=7$, then what is the value of $g(1)$? I tried doing integrating by parts repeatedly, but no success. Wolfram Alpha also ...
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0answers
11 views

On the properties of extended integrals

Suppose that {$A_1,A_2,...$} is a countable collection of subset of $\mathbb{R}^n$ such that $A_i\cap A_j=\emptyset$, for $i\ne j$. Put $A=\bigcup \left \{ A_i \right \}$. Let $A$ be open. I will use ...
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0answers
49 views

Integral with complicated exponentials [on hold]

I have an integral like: \begin{equation} \int_{0}^{\tau}\frac{1}{(1 + ax^{2})^{2}}e^{-\frac{bx^{2}}{(1 + ax^{2})}}dx \end{equation} where $a$ and $b$ are constants. Does somebody have any idea on ...
0
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2answers
63 views

How can I examine if this limit exists?

I can't integrate $e^t/t$ so I don't really know how to examine if this limit exists: $$ \lim_{x\to\infty} e^{-x} \int_1^x \frac{e^t}{t} dt $$ Thanks for any help.
0
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0answers
23 views

Differentiation of Laplace Transform

It is known that The $s-$derivative rule states that $$ \mathcal{L} (t^{n} f) = (-1)^{n} F^{(n)} (s) $$ The proof for the laplace differentiation involves \begin{align*} F'(s) &= ...
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1answer
38 views

Solve $I=\int_0^1\frac{ln(1+x)}{1+x^2}dx$ [duplicate]

Solve $$I=\int_0^1\frac{ln(1+x)}{1+x^2}dx.$$ After let $x=\tan t$, $I=\int_0^{\pi/4}ln(1+\tan t)dt$ and I stuck here.
0
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0answers
32 views

Does U-substitution only work in multiplication? [on hold]

Apart from functions in multiplication, I have also just seen how the U-substitution can work in division as well: $\int {x \over (x+2)^{1/4}}$ Here, you can put $x +2 = u$ and solve it. Is it ...
2
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0answers
43 views

Problem with an integral involving hypergeometric functions

I want to evaluate the integral \begin{align} K=&\int\limits_{0}^{1}\mathrm{d}z\:(1-z)^{-2\varepsilon}z^{-\varepsilon} \int\limits_{0}^{1}\mathrm{d}t\:(1-t)^{-1-\varepsilon}t^{-\varepsilon} ...
2
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0answers
50 views

What does this formula mean? [on hold]

What is this formula about? Does it make sense? I'm serious. I like this, but feel foolish when I'm asked what it does mean ;-)
2
votes
2answers
114 views

Compute $\int\frac{1}{3+\cos^3{x}}\mathrm{d} x$

I have an integral which seems hard for me: $$\int\frac{1}{3+\cos^3{x}}\,\mathrm{d}x.$$ If I use Weierstrass substitution I get $I=\int{\frac{(1+t^2)^2}{3(1+t^2)^3+(1-t^2)^3}dt} $ I was stuck here
4
votes
2answers
45 views

Different results in integrating both sides of $\sin{2x}=2\cos x\sin x$

I feel like there is something I am missing here. When integrating both sides of the trigonometric identity $\sin{2x}=2\cos x\sin x$ I get different results. The left side of course results in ...
0
votes
1answer
28 views

Rational Funtion Integration

This looks to be a simple problem, but it has me stumped. I already have the answer, but a step-by-step solution would be appreciated. $$\int\frac{x+4}{x^2+2x+5}$$
1
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3answers
26 views

Riemann integral on trigonometric functions

I have to calculate Riemann integral of function $g:[0;\pi/4]\rightarrow\mathbb{R}$ (on interval $[0;\pi/4]$) given as $g(x)=\frac{\tan(x)}{(\cos(x)^2-4)}$. Function $g$ is continous on interval ...
2
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0answers
12 views

Question about an integral involving a miminum in the integrand

Say I have an integral of this form: $\int_{0}^{min(x,y)}f(x,u)f(y,u)du$ and I want to get it in this form: $\int_{0}^{y}f(y,u)du\int_{0}^{x}f(x,u)du$, does anyone know of any practical way to do ...
0
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1answer
31 views

Integration with vectors

I am trying to solve below integration $$\int_{0}^{\infty}\hat{k}\frac{e^{ikR}}{k-l}dk$$ here $R,l$ are constants and $\hat{k}$ is a unit vector of $\textbf{k}$. And as usual ...
2
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0answers
69 views

Connection between Dirichlet series and integration?

For quiet sometime I've been working on an idea of mine: Basis We define the following basis: $$ A_n= ( \underbrace{00000...}_{n-1\text{ times}} 1 )^T $$ Hence, $$ A_1 =(111111 ... )^T $$ $$ A_2 ...
0
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2answers
26 views

Evaluate convolution integral

Can someone tell me if I am calculating this integral correctly.
0
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1answer
37 views

Calculate the flux through a surface S from a field described by vectors

I have encountered yet another example which is not that typical. I need to calculate: $$\iint\limits_{S} \vec{F} \vec{ds} =\text{ ?}$$ Where the $F$ and $S$ are as follows ($S$ is oriented ...
0
votes
1answer
12 views

Convolution integral involving two Heaviside functions

I am having trouble solving the following integral involving two Heaviside functions, obtained from a Laplace transform convolution: $\Large \int_0^t \frac{\tau}{\sqrt{\tau^2 - \alpha^2}} H(\tau - ...
5
votes
2answers
47 views

How to find the integral $\int_0^z \exp(ax)x^{b-1}(1-x)^{c-1}dx$?

How to find the integral $\int_0^z \exp(ax)x^{b-1}(1-x)^{c-1}\text{d}x$ where $b,c\in \mathbb{C}, \Re(b)>0, \Re(c)>0$?
6
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4answers
250 views

Problem of Integration by Parts involving algebraic and exponential functions

Can anyone please help me in solving this integration problem $\int \frac{e^x}{1+ x^2}dx \, $? Actually, I am getting stuck at one point while solving this problem via integration by parts.
1
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1answer
26 views

Upper bounding incomplete gamma function

For $0<\theta, \lambda<1$ and $c>1$, we wish to upper bound the following gamma function: $$\int_{\theta}^{1} t\exp \left(-c\left(\lambda t+\frac{1}{t}\right) \right)dt$$
0
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3answers
69 views

Integration $ \int x^2 \cos(nx) dx $

How do I solve this integral? $ \int x^2 \cdot \cos(\frac12 n\pi x)dx $ I tried integration by parts... $ u=x^2\\u'=\frac13 x^3\\v'=\cos(\frac12 \pi n x)\\v=sin(\frac12 \pi n x)\cdot \frac12 \pi n ...
0
votes
2answers
22 views

Integral differentiation with infinite bound (differentiation of expected value)

I am trying to proove the following: $$\frac{d}{dx}\int_x^{\infty}(z-x)f_Z(z)dz=1-F_Z(x)$$ Where $f_Z$ and $F_Z$ are resp. the probability density and cumulative distribution functions of a random ...
3
votes
1answer
60 views

On the integration of a Lebesgue measurable function

Consider a function $f$ defined as $f:[0,2\pi]\to \mathbb{R}$ such that $\begin{equation} f(x)=\inf_{n\in \mathcal{N}} \sin^2 (2^n x) \end {equation}$ Is possible to give a decent bound of ...
0
votes
1answer
78 views

Integral $\frac{\sqrt{1+x^2}}{x}$

I understand that I will be using trig substitution, and tangent will be what is used, but I get confused later down the road when integrating with the trig.