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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1
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3answers
20 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
10 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
24 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
23 views

Connection between Dirichlet series and integration?

For quiet sometime I've been working on an idea of mine (I think its not discovered at least, within my googling capabilities). I think I found a connection between the Dirichlet series and ...
0
votes
2answers
22 views

Evaluate convolution integral

Can someone tell me if I am calculating this integral correctly.
0
votes
1answer
11 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
36 views

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

How to find the integral $\int_0^z exp(ax)x^{b-1}(1-x)^{c-1}dx,~b,c\in \mathbb{C}, Re(b)>0, Re(c)>0$?
4
votes
3answers
206 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
18 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
61 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
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2answers
18 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 ...
-1
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0answers
25 views

On the integration of a Lebague measurable function [on hold]

Sincerely need help on this question, anyone who has any ideas please don't hesitate to tell me :)
0
votes
2answers
72 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
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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
40 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
21 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
34 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
20 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
23 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
66 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
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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
23 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
32 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
93 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
56 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
20 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
126 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
37 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
26 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
89 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
96 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
70 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.
1
vote
0answers
22 views

Cauchy formula for repeated Lebesgue integration

Is there an equivalent of the Cauchy formula for repeated integration (https://en.wikipedia.org/wiki/Cauchy_formula_for_repeated_integration) for the following \begin{equation} f^{(-n)}(x) = \int_a^x ...
2
votes
1answer
34 views

How to determine the function from the following?

The graph of a certain function contains the point $ (0,2)$ and has the property that for each number 'p' the line tangent to $y = f(x)$ at $(p, f(p))$ intersect the x-axis at p + 2. Find $f(x)$ The ...
1
vote
1answer
35 views

Evaluate the integral $\int_0^{\infty} e^{\frac{-t(s-1)^2}{2}} \left( \frac{t(s-1)^3}{3} \right) ds$

I am attempting to evaluate the integral (where $t \rightarrow \infty$) $$I(t) = \int_0^{\infty} e^{\frac{-t(s-1)^2}{2}} \left( \frac{t(s-1)^3}{3} \right) ds$$ which occurs in the calculation of the ...
1
vote
2answers
37 views

“Trig Substitutions”, I tried half- angle and trig indentity in this one, but doesn't work

I´m really lost in this one. $\int \sin^3 (2x) \cos^2 (2x) dx$ I know that the answer is: $\frac{1}{10}cos^5(2x)-\frac{1}{6}cos^3(2x) + c$ Please help
4
votes
3answers
98 views

Why does WolframAlpha's expression for $\int\frac{dx}{x\sqrt{x^4-4}}$ disagree with my own?

$$\int\frac{1}{x\sqrt{x^4-4}}$$ My teacher gave us these notes and I'm unsure if they're correct. Wolfram gives a different answer, and when I derive I might have messed up. Thanks.
1
vote
3answers
28 views

integration by parts of $25\, (1-\sin^{2}x)$

I need help solving this integration of parts problem. I've tried a few different solutions and keep getting the wrong answer. This question is in regards to this problem take the integral by parts ...
2
votes
1answer
22 views

Integrating a cost function over a normal distribution

Let's say you have a cost function $C(x)$ and you want to understand the expected cost if the input follows the normal distribution $$X \sim \mathcal{N}(\mu,\sigma ^2)\\ $$ If I want to find my ...