Questions tagged [integration]
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 the integrals being considered. This tag often goes along with the (calculus) tag.
60,970
questions
1
vote
0answers
18 views
Show that $f=\sin(x)$ if $x\in\mathbb{Q}$ and $f(x)=\cos(x)$ if $x\in\mathbb{R}\setminus\mathbb{Q}$ is not Riemann integrable on $[0,1]$
Show that $f=\sin(x)$ if $x\in\mathbb{Q}$ and $f(x)=\cos(x)$ if $x\in\mathbb{R}\setminus\mathbb{Q}$ is not integrable on $[0,1]$. I know that there is already a post on this function, but I didn't ...
0
votes
2answers
70 views
Evaluate the integral $\int_{\mathbb R} \frac{\exp\left(-y(x^2+z\cdot x\sqrt{x^2+1} ) \right)}{\sqrt{x^2+1}}\mathrm dx$
I try to calculate the integral
\begin{align}
f(y,z):=\int_{\mathbb R} \frac{\exp\left(-y(x^2+z\cdot x\sqrt{x^2+1} ) \right)}{\sqrt{x^2+1}}\mathrm dx
\end{align}
on the set $y>0$ and $0<z<1$. ...
2
votes
1answer
1k views
Show $f$ is integrable in each subinterval, $[x_{i−1}, x_i]$, and further, $\int_a^bfdx=\sum\limits_{x_{k-1}}^{x_k}\int\limits_{x_{k-1}}^{x_k}fdx$.
Let $f$ be integrable on $[a,b]$, and let Let $P = \{x_0,x_1,x_2,...,x_{n−1},x_n\}$ be any partition of $[a,b]$. Show that $f$ is integrable in each subinterval, $[x_{i−1}, x_i]$, and further, $\int_a^...
0
votes
1answer
16 views
Dependance of variables in variational calculus
I stumbled across the following functional for which a stationary path should be found:
$$ L[y]=\int\limits_a^b x[y‘(x)]^4+ ... \,\mathrm{d}x$$
Where the dots indicate the other terms that were ...
3
votes
0answers
43 views
Let $f(x)=\frac{1}{n}$ if $x=\frac{1}{n}$ and $f(x)=0$ if not. Show that $f$ is Riemann integrable
Let $f(x)=\frac{1}{n}$ if $x=\frac{1}{n}$ and $f(x)=0$ if not. Show that $f$ is Riemann integrable on $[0,1]$. I would like to have a feedback on my proof and to know if it holds.
My attempt is to ...
0
votes
1answer
91 views
Convergence integral Riemann zeta
In the paper of Riemann and the book of Edwards I encountered the following representation for $\zeta$:
$$
\int_0^\infty\frac{x^{s-1}}{e^x-1}\,dx=\Pi(s-1)\zeta(s)
$$
which due to the fast growth of $e^...
0
votes
1answer
12 views
On the integrals of two complex valued functions which get closer
Suppose $f,g$ are complex valued continuous functions defined on positive real axis. Suppose for all $\epsilon>0$, there is $M$ such that $|f(x)-g(x)|<\epsilon$ for $x>M$ and it is given ...
1
vote
1answer
50 views
How to obtain $\frac{\pi^2}{6}$ from $\displaystyle\int_0^1 \frac{-\ln(1-x)}{x} \, dx $
\begin{align}
& \int_0^1 \frac{-\ln(1-x)}{x} \, dx = \int_0^1 1+\frac{x}{2}+\frac{x^2}{3} + \frac{x^3}{4}+\cdots \, dx \\[8pt]
= {} &\left[1+\frac{x}{2}+\frac{x^2}{3}+\frac{x^3}{4} + \cdots \...
1
vote
3answers
35 views
Let $a$ be a non zero real number. Evaluate the integral $\int \frac{-7x}{x^{4}-a^{4}}dx$
I hit a wall on this question. Below are my steps
$$\int \frac{-7x}{x^{4}-a^{4}}dx=-7\int \frac{x}{x^{4}-a^{4}}dx$$
Let $u=\frac{x^2}{2}, dx = \frac{du}{x}, x^{4}=4u^{2}.$
$$-7\int \frac{1}{4u^{2}-a^{...
0
votes
1answer
34 views
How to approximately calculate the integration : $ I = \int_{0}^{\infty}(\frac{1}{x^2+a^2})(\frac{1}{(x-1)^2+b^2}) dx $
Integration :
$ I = \int_{0}^{\infty}(\frac{1}{x^2+a^2})(\frac{1}{(x-1)^2+b^2}) dx $
when : (i) $a\ll 1$, $b\sim 1$;
(ii) $a=b\gg 1$
I want some hints for the transformation of the parameters or ...
0
votes
1answer
23 views
Let $f(x)=2|x|+1$ if $x\in\mathbf{Q}$ and $f(x)=0$ if not. Show that $f$ is not Riemann integrable
Let $f(x)=2|x|+1$ if $x\in\mathbf{Q}$ and $f(x)=0$ if $x\in \mathbf{R}/\mathbf{Q}$. Show that $f$ is not Riemann integrable on $[-2,3]$. I would like to have a feedback on my proof and to know if it ...
6
votes
0answers
77 views
On the integral $\int_0^1\frac{\arctan\sqrt{t^2+a}}{(t^2+b)\sqrt{t^2+a}}dt$
Let $0<b<a$ and define $$J(a,b)=\int_0^1\frac{\arctan\sqrt{t^2+a}}{(t^2+b)\sqrt{t^2+a}}dt.\tag1$$
I am seeking a closed form for $J(a,b)$.
I was motivated to find a closed form for $(1)$ after ...
0
votes
1answer
27 views
How to calculate the maximum volume of a conical frustum?
This is just a question for my research on the ideal cup size to minimize the amount of plastic used.
Assuming that the conical frustum is in a shape of a cup. How do I go about in solving the maximum ...
13
votes
2answers
341 views
+100
How to Evaluate $\int_{0}^{1} \frac{x^2 \ln(1-x^4)}{1+x^4}$?
How to evaluate
$$ \int_{0}^{1} \frac{x^2 \ln(1-x^4)}{1+x^4} \,dx \approx -0.162858 \tag{1}$$
The integral arises in the computation of
$$\left( \sum_{n=1}^{\infty} \frac{(-1)^n}{n}\right)\left(\sum_{...
0
votes
1answer
62 views
Fitting a derivative of a curve to data
Some derivatives cannot be integrated to an analytic function but have the function embedded in the derivative itself. I need to know how to fit those derivative functions to a set of data. Here is ...
0
votes
0answers
16 views
Integral over factorials/reciprocal beta function / generatnig function of $\ln\left({\ln(1+c)\over c}\right)$
I know from the product representation of $\ln(1+c)/c$ that the following:
$$\ln\left({\ln(1+c)\over c}\right)=\sum_{n=1}^{\infty}\int_{-1}^0 \frac{(n+x)!}{n! x! n}c^n (-1)^n dx $$
I wondered what ...
0
votes
0answers
14 views
set the limits of integration in the triple integral [closed]
set the limits of integration in the triple integral ∭f(x,y,z)dxdydz if the area V is limited to the indicated surfaces.draw the region of integration
0
votes
0answers
9 views
Is the trapezoidal rule equivalent to Gauss-Chebyslev quadrature?
I encountered a paper in which the author derives a quadrature rule for integration of a function $f(x)$ over a domain $[0,L]$, $$\int_0^L f(x) dx = \sum_{i=1}^{N-1} w_i f(x_i).$$ I'll give a quick ...
0
votes
2answers
74 views
Why this integral is $0$? $ \int_{0}^{2 \pi} \frac{2 \cos(2t) (\sqrt{2}ie^{it})}{4e^{i4t}-2\sqrt{2}e^{i2t}+1}dt$
I was unable to calculate this integral directly $$ \int_{0}^{2 \pi} \frac{2 \cos(2t) (\sqrt{2}ie^{it})}{4e^{i4t}-2\sqrt{2}e^{i2t}+1}dt$$
so I put it into wolfram and it says that it is equal to zero. ...
0
votes
0answers
53 views
Solve this integral $\int_0^9x^9\sqrt{x^3 + 2}dx$
$I = \displaystyle\int\limits_0^9x^9\sqrt{x^3 + 2}dx$
I let $u = \sqrt{x^3 +2}$
Then $I = \dfrac23\displaystyle\int\limits_{\sqrt2}^{\sqrt{731}}u^2(u^2 -2)^2\sqrt[3]{u^2 -2}du$
After that, I let $t = ...
1
vote
2answers
72 views
show that $f$ is Riemann integrable on $[0,1]$
Let $f(x)=\sin\left(\frac{1}{x}\right)$ if $0<x\le1$ and $f(x)=0$ if $x=0$. Show that $f$ is Riemann integrable on $[0,1]$ and calculate it's integral on $[0,1]$.
I would like to know if my proof ...
9
votes
4answers
196 views
Prove $ \int_{5\pi/36}^{7\pi/36} \ln (\cot t )dt +\int_{\pi/36}^{3\pi/36} \ln (\cot t )dt = \frac49G $
I discovered this integral heuristicaly
$$ \int_{\frac{5\pi}{36}}^{\frac{7\pi}{36}} \ln (\cot t )\>dt +\int_{\frac{\pi}{36}}^{\frac{3\pi}{36}} \ln (\cot t )\>dt = \frac49G $$
where $G$ is the ...
-2
votes
0answers
13 views
Finding fourier series of odd square wave
$$f(t) = -1$$
$$\frac{-T}{2} \ne t<0$$
$$+1, 0<=t<\frac{T}{2}$$
Since it's an odd wave, coefficient involving cos will be zero. I'm trying to calculate coefficient involving sine
$b_r = \frac{...
0
votes
4answers
589 views
Using integral to find the area under a portion of a semicircle
I've done quite a bit of research, but I always fail right before the end.
I know the integral of a circle, I know the substitution $$\theta = \arcsin\left(\frac x r\right)$$ However, I never get ...
0
votes
1answer
72 views
How to integrate $\sin{\theta}dt$ [closed]
I was doing some physics and I got on a problem which is how to integrate $\sin{\theta}dt$, but the problem is that $\theta$ is a function in terms of $t$ and we don't know what that function is.
I ...
0
votes
2answers
47 views
How to prove (using integration by parts)
Could you help to prove this:
$$Y(\lambda) = \int_{a}^{b} g(x) e^{i \lambda f(x)} \,dx$$
we should use integration by parts to get
$$Y(\lambda) = \frac{1}{i \lambda }e^{i \lambda f(x)} \frac{g(x)}{...
248
votes
5answers
20k views
Evaluate $ \int_{0}^{\frac{\pi}2}\frac1{(1+x^2)(1+\tan x)}\:\mathrm dx$
Evaluate the following integral
$$
\tag1\int_{0}^{\frac{\pi}{2}}\frac1{(1+x^2)(1+\tan x)}\,\mathrm dx
$$
My Attempt:
Letting $x=\frac{\pi}{2}-x$ and using the property that
$$
\int_{0}^{a}f(x)\,\...
1
vote
1answer
13 views
Weak derivative of $|x|$ not in $W^{1,p}(-1, 1)$, another possible solution
I'm referring to the question presented here: Weak derivative of $u(x)=|x|$ not belong in $W^{1,p}(-1,1)$
In short, it is easy to see that the weak derivative of $|x|$ (defined on $(-1, 1)$) is the ...
1
vote
1answer
19 views
Matrix change of basis, versus vector
I am under the impression, that changing the basis of a vector involves multiplication by a matrix:
$$
\mathbf{x}'=\mathbf{A}\mathbf{x}
$$
where $\mathbf{x}',\mathbf{x}\in \mathbb{R}^n$ and $\mathbf{A}...
0
votes
1answer
50 views
Find $\int \frac{e^{-x}\sin x}{x} \ dx$.
I'm trying to compute $\int \frac{e^{-x}\sin x}{x} \ dx$. This resembles integrating $\int e^{-x}\sin x \ dx$ so my main thought about how to approach this is by finding some connection with that. I ...
1
vote
4answers
53 views
Find $\int \frac 1 {(x^2 +a^2)^2} dx $.
I am trying to integrate $\int \frac 1 {(x^2+a^2)^2} \ dx$. The only thing that I can think to try is substitution, $u=x^2+a^2$ so that $\frac{du}{dx}=2x \Rightarrow du = 2x\ dx = 2\sqrt{u-a^2}\ du$ ...
2
votes
1answer
43 views
Show that $f(x)=x$ if $x\in\mathbf{Q}$ and $f(x)=1-x$ if $x \in\mathbf{R}/\mathbf{Q}$ is Riemann integrable on $[0,1]$
Let $f(x)=x$ if $x\in\mathbf{Q}$ and $f(x)=1-x$ if $x \in\mathbf{R}/\mathbf{Q}$. Show that $f$ is Riemann integrable on $[0,1]$.
I know that there are a few posts on this function, but I didn't see ...
1
vote
1answer
51 views
Is there a definition of Riemann integration in $\mathbb{R^n}$?
I know that Riemann integration is well-defined for a function $f: \mathbb{R} \to \mathbb{R}$.
I would like to ask whether there is a definition of Riemann integration for a function $f: \mathbb{R^n} \...
2
votes
1answer
21 views
Triple Integral in Cylindrical Coordinates using x axis instead of z axis
Today in my Calculus class my teacher made an example using change of variables using this problem:
Find the volume of the solid inside the cylinder $y^{2} + z^{2} = 2y$ bounded by $x=0$ and the ...
7
votes
3answers
180 views
Is this integral $\int_0^\infty\frac{\cos(a x+ 2b \arctan x)}{x^2+1}dx$ exactly zero when $b\in\mathbb{N}$?
I recently encountered this integral
$$\int_0^\infty\frac{\cos(a x+ 2b \arctan x)}{x^2+1}dx$$
which suspiciously close to 0 for nonzero integer values of $b$, as indicated by numerical calculations. ...
0
votes
1answer
48 views
Do I have to use integral for this problem? [closed]
Find the area of the shape formed by the $(x+y)^5=2021x^2y^2$ function graph.
2
votes
2answers
49 views
Using Calculus to Solve for a Differential Equation
The question I have is: $x''(t)+yx'(t)+z^2x(t)=0$, where y and z $\in \mathbb{R}$ without $0$. $t \geq 0$ by the way. The total energy of the system is $$\frac{1}{2}(x'(t))^2+ \frac{1}{2}z^2x(t)^2$$ ...
12
votes
4answers
432 views
Definite integral - closed form: $\int_{0}^{\infty}\cos\left(x^{4} + 1 \over x^{2}\right)\,{\rm d}x$
I'm struggling with this definite integral:
$$
\int_{0}^{\infty}\cos\left(x^{4} + 1 \over x^{2}\right)\,{\rm d}x.
$$
Any help will be greatly appreciated.
1
vote
1answer
53 views
Integral notation in Shifrin's *Multivariable Mathematics*
On page 348 of Shifrin's Multivariable Mathematics, for a 1-form $\omega=\sum F_{i}dx_{i}$ on $\mathbb{R}^{n}$ and a parameterized curve $C$ given by a function $\mathbf{g}:[a,b]\rightarrow\mathbb{R}^{...
-1
votes
0answers
35 views
Proving mean value theorem without Rolle's theorem
I started to wonder whether I could prove the mean value theorem without using Rolle's theorem by using limits and integration. My point was to see if I could prove the theorem by noting that it ...
0
votes
1answer
36 views
Limit of an definite integral
Calculate
$$\lim\limits_{n\to\infty}\int_0^1 \frac{x^n(ax^2+ax+1)}{e^x}.$$
a) $0$
b) $a$
c) $2a+1$
d) $\dfrac{2a+1}{e}$
I tried to note separated $\displaystyle\int_0^1 \frac{x^n\cdot x^2}{e^x}$ with ...
2
votes
1answer
49 views
Need help with proof on trigonometric integrals
Question:
Let $f:[0,\pi]\rightarrow\mathbb{R}$ be a continuous function. Show that, if $$\int\limits_0^{\pi} f(t)\sin(t)dt = \int\limits_0^{\pi} f(t)\cos(t)dt =0,$$ then the equation $f(x)=0$ admits ...
1
vote
2answers
13 views
Solving differential equation with boundary conditions $y=2$ at $x=0$
I need to find a solution to this differential equation $(1+x^2) $$\tfrac{\mathrm{d}y}{\mathrm{d}x} = x-xy^2$, with boundary condition $y=2$ at $x=0$
I collect all terms on the RHS to get $$\tfrac{\...
7
votes
2answers
603 views
How to integrate$\int_0^1 \frac{\ln x}{x-1}dx$ without power series expansion
I happen to watch the video here,
which gives a solution to the definite integral below using the power series approach. Then answer is $\frac{\pi^2}{6}$, given by:
$$\int_0^1 \frac{\ln x}{x-1}dx=\...
8
votes
1answer
319 views
Evaluate $\int _0^{\infty }\frac{\ln \left(x^5+1\right)}{x^2+1}\:dx$ with real methods
I started like this:
$$\int _0^{\infty }\frac{\ln \left(x^5+1\right)}{x^2+1}\:dx\:
=\int _0^1\frac{\ln \left(x^5+1\right)}{x^2+1}\:dx+\overset {x=\frac{1}{x}}{\int _1^{\infty }\frac{\ln \left(x^5+1\...
3
votes
1answer
175 views
An interesting ODE
I've been thinking about approaches to solve
$$\frac{\textrm{d}^{3}y}{\textrm{d}x^{3}}=\textrm{e}^{-y(x)}.$$
My initial thought was to set $y(x)=\ln(z(x))$ in order to obtain an ODE relating $z$ to $x$...
-1
votes
0answers
18 views
Evaluating $ \int_0^{\Lambda}r^{d - 1}\log\left(1 + a\sqrt{r^2 + m_1^2} + b\sqrt{r^2 + m_2^2}\right)\,dr$ [closed]
Does anyone knows how to do the following integral?
\begin{equation}
I = \int_0^{\Lambda}r^{d - 1}\log\left(1 + a\sqrt{r^2 + m_1^2} + b\sqrt{r^2 + m_2^2}\right)\;dr
\end{equation}
Here the constants ...
-1
votes
4answers
63 views
Find $c\in\mathbb{R}$, and $f$ such that $\int_c^xt\cdot f(t)\,\mathrm{d}x=x\sin (x)+\cos (x)+\frac{x^3}3$
Find a number and a function that satisfies:
$$\displaystyle\int_{c}^{x}t\cdot f(t)\,\mathrm{d}x=x\sin (x)+\cos (x)+\dfrac{x^3}{3}$$
I tried taking derivative both sides so I got:
$$\left(\...
0
votes
0answers
30 views
Airy functions - integral
I am struggling to simplify following integrals consisting of following Airy functions.
$u=C_3 e^{-\beta_0 x_2} \int _0^{x_2} e^{2\beta_0 x_2} \int _0^{x_2} e^{-\beta_0 x_2} Ai(-\frac{(-i A Re \...
0
votes
2answers
77 views
Integrability of Thomae function and value of its integral
I am trying to solve the following exercise (from Axler's Measure, Integration and Real Analysis) and I would like to have some help in finishing my proof.
"Define $f:[0,1]\to\mathbb{R}$ as ...
