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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0
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1answer
25 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
votes
0answers
24 views

Does U-substitution only work in multiplication?

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
votes
0answers
29 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} ...
0
votes
0answers
40 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 ;-)
0
votes
1answer
62 views

Compute $\int\frac{1}{3+\cos^3{x}}\mathrm{d} x$ [on hold]

I have an integral which seems hard for me: $$\int\frac{1}{3+\cos^3{x}}\,\mathrm{d}x.$$
4
votes
2answers
42 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
vote
3answers
23 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
votes
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
votes
1answer
27 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 ...
0
votes
0answers
38 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: $$ A_n= ( \underbrace{00000...}_{n-1 times} 1 )^T $$ Hence, $$ A_1 =(111111 ... )^T $$ $$ A_2 = ...
0
votes
2answers
25 views

Evaluate convolution integral

Can someone tell me if I am calculating this integral correctly.
0
votes
1answer
22 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
11 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 - ...
3
votes
1answer
39 views

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

How to find the integral $\int_0^z exp(ax)x^{b-1}(1-x)^{c-1}dx,~b,c\in \mathbb{C}, Re(b)>0, Re(c)>0$?
5
votes
4answers
238 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.
0
votes
1answer
23 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
votes
3answers
65 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
20 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
54 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
77 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.
1
vote
0answers
17 views

When is the riemann integral of a finite-variation process square-integrable?

Let us say $X_t$ is a continuous finite-variation process and $f(t, x)$ a $C^{1, 1}$ function. We define $$ Y_t = \int^t_0 f(s, X_s)\, \mathrm ds $$ Are there any general results pertaining to ...
3
votes
2answers
43 views

Trouble solving this differential equation: $x'=3(x-2)$, $x(0)=-1$.

Find the solution of the differential equation x'=3(x-2) given initial value condition of x(0)=-1 Here's my attempt. x'=3(x-2) dx/dt = 3(x-2) dx/x-2 = 3dt int dx/x-2 = int 3dt+c ln|x-2| = 3 + C ...
1
vote
1answer
23 views

Divergence theorem and applying cylindrical coordinates

This time my question is based on this example Divergence theorem I wanted to change the solution proposed by Omnomnomnom to cylindrical coordinates. $$ \iiint_R \nabla \cdot F(x,y,z)\,dz\,dy\,dx = ...
0
votes
0answers
33 views

Real integral done by complex methods [duplicate]

$\int_{-\infty}^{\infty} \frac{cosx}{x^2+25} dx $ = $ \frac{\pi}{5e^5}$ Any ideas?
5
votes
1answer
38 views

green's second identity application

I need to use the green's second identity in order to prove the following equality: $$ \int_{\mathbb{R}^2} \ln (\sqrt{x^2+y^2})\Delta f = -2\pi f(0)$$ where $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ ...
1
vote
0answers
21 views

Bound on the mean value of function involving Hilbert transform

Consider the integral $$\int_{-\infty}^{\infty} x|A|^2_x\mathbb{H}(|A|^2_x) \ dx,$$ where $A=A(x,t)$ is a complex valued, compact function (I mean this in the heuristic sense that $A$ vanishes ...
0
votes
1answer
24 views

Volume of revolution on an area crossing the axis

Suppose a question asks for the volume of revolution about the x axis to be found on a piece of area enclosed between 2 graphs, where the area crosses the x-axis. In this case, the method involving ...
0
votes
0answers
22 views

Switching the order of integration [duplicate]

I need to switch the order of integration for the following function: $$ \int_{0}^{1} \int_{0}^{1-x} \int_{0}^{x+y} f[x,y,z] dz dy dx $$ to the order: $$ \int_{?}^{?} \int_{?}^{?} \int_{?}^{?} ...
0
votes
2answers
67 views

How to solve this integration?

I want to solve this $$\int_0^w (b/x)^{a+1} e^{(cx-(b/x)^a)} dx$$ where $a$, $b$, and $c$ are arbitrary positive real numbers. Do i have to solve it numerically? I have no clue to solve this ...
0
votes
0answers
27 views

About a curve and its direction at every point

Consider the following class of problems: Consider a plane (or $\mathbb{R}^n$ in general). Let to any point of the plane corresponds a set of directions (set of unit vectors, I mean). Does there ...
1
vote
1answer
25 views

Finding the root mean square of a sum of trig functions

$$v(t) = 3 - 2\sin(t) + 8\sin^2(t)$$ To find the rms of this function, I first figured out that the period $T = 2\pi$. I then set up the equation: $V = \sqrt{\frac{1}{T}\int^T_0v^2(t)\,dt}$ ...
0
votes
0answers
26 views

HJM Model vs Leibniz integral rule

I state that I'm an electronic engineer (undergraduate), then the my knowledges about advanced mathematics are almost null. A colleague asked to me an help about one point of the proof of the theorem ...
0
votes
2answers
34 views

Which is the justification for this indefinite integral relation? [on hold]

Why is the following indefinite integral equation correct: $$ \int \frac{\cot(x)}{\sin^2(x)} dx= -\frac{1}{2}\cot^2(x) $$ What are the necessary steps?
-3
votes
1answer
40 views

Find area under the given curve [on hold]

Find the area under the curve $$\displaystyle y = \frac{|x-3| + |x+1|}{|x+3| + |x-1|},$$ the $x$-axis , $x = -3$ and $x = 1.$
0
votes
1answer
99 views

How to evaluate $\int \dfrac {x^3} {1+x^6} dx $?

How to evaluate $\int \dfrac {x^3} {1+x^6} dx $ ? I am completely at a loss , please help , thanks in advance .
2
votes
0answers
58 views

solving definite integral problems without complex line integral

It is well known that some definite integrals such as $$\int_{0}^{\infty} \frac{dx}{a+\cos{x}}$$ $$\int_{0}^{\infty} \frac{\sin{x}}{x}dx$$ are solved by using complex analysis techniques. (It uses ...
4
votes
4answers
57 views

I'm stuck in this one of trig substitution for fuctions.

I got this: $$\int\frac{dx}{\sqrt{(4x^2-9)^3}}.$$ I know that the answer is: $$\frac{x}{9*\sqrt{4x^2-9}}+c.$$ And with the steps that I know about this type of substitution, I came up here, but.. ...
-5
votes
0answers
21 views

What method we use when n = odd for evaluate the integral using simpson's rule ??? [on hold]

hi any one can tell me What method we use when n = odd for evaluate the integral using simpson's rule ??? plz help....
6
votes
2answers
133 views

Evaluating $~\int_0^1\sqrt{\frac{1+x^n}{1-x^n}}~dx~$ and $~\int_0^1\sqrt[n]{\frac{1+x^2}{1-x^2}}~dx$

How could we prove that $$\int_0^1\sqrt{\frac{1+x^n}{1-x^n}}~dx~=~a\cdot2^{a-1}~\bigg[\frac12~B\bigg(\frac a2,~\frac a2\bigg)~+~B\bigg(\dfrac{a+1}2,~\dfrac{a+1}2\bigg)\bigg],$$ where ...
0
votes
1answer
38 views

Can you explain the result in this one, please?

I tried complete the square, but it doesn't work I got this: $\int\frac{xdx}{\sqrt{3-2x-x^2}}$ And I know that the answer is: $-{\sqrt{3-2x-x^2}}-\arcsin(\frac{x+1}{2})+c$
7
votes
2answers
68 views

Summation of the reciprocals of the product of consecutive integers

It is well known that there is a closed formula for: $$\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \cdots + \frac{1}{(n)(n + 1)}$$ And likewise for: $$\frac{1}{1 \cdot 2 \cdot 3} + \frac{1}{2 \cdot ...
0
votes
2answers
27 views

Example of a Riemann integrable sequence of functions such that the the sequence of Riemann integrals diverges but… (see below)?

Is there a sequence $(f_n)$ of Riemann integrable functions such that $\lim f_n(x) = f(x)$ almost everywhere on $[a,b]$ and $\lim\int_a^bf_n$ does not exists in Riemann sense, but it does in Lebesgue ...
0
votes
1answer
10 views

Solving Poisson's Equation in 1-D for a point charge?

Ok so I was trying to solve the Poisson's equation for a point charge with a Fourier transform to get the familiar equation. This is what I did so far: So ultimately I am trying to solve this in 3 ...
4
votes
1answer
102 views

for which values of $\alpha \in \mathbb R$ is $f$ integrable?

For which values $\alpha \in \mathbb{R}$ is $f$ integrable? $$f: \mathbb{R}^2 \rightarrow \mathbb{R} : f(x, y) = x \frac{\ln(1 + x^2 + y^2)}{(x^2 + y^2)^\alpha} $$ if $ (x,y) \neq (0, 0) $ and $ ...
1
vote
2answers
37 views

Limit of an integral of a continuous real-valued function

If $f:[0,{\infty})\to\mathbb R$ continuous and $\lim_{x\to\infty} f(x)=a$. Show that: $$ \lim_{x\to\infty} \frac1x\int_{0}^{x} f(t)\ \mathsf dt = a. $$ If: $$ \lim_{x\to\infty} \frac1x ...
1
vote
4answers
89 views

How does $\int (\cos(x))^{-2}dx$ equal to $\tan(x)$?

How does $$\int \frac{1}{\cos^2(x)} dx= \tan(x)+ C$$ ?
4
votes
5answers
98 views

How do I integrate$ \int\frac{1}{e^{2x}+e^x} \,dx $ [on hold]

How do I integrate following function? $$ \int\frac{1}{e^{2x}+e^x} \,dx $$
2
votes
1answer
69 views

Finding a general integral

$$ \int\limits_{0}^{1}{\frac{\ln(1+{t}^{a})}{1+t} \;\mathrm{d}t} $$ I have tried many tings but I am just not successful in any of them - Feynman, summation inside integral, Beta function ...
-1
votes
5answers
71 views

Integration of $\displaystyle \frac{e^x-1}{e^x+1}$ w.r.t. $x$ [on hold]

I tried a lot but unable to find out the solution of $$ \int\left(\frac{e^x-1}{e^x+1}\right)dx$$ Please solve it.