Questions tagged [definite-integrals]
Questions about the evaluation of specific definite integrals.
0
votes
0answers
15 views
The integral of $1/r_{12}$ in the cartesian space.
So I have two negative point particles $p$ and $p'$, with positions ${\vec x} = (x,y,z)$ and ${\vec x}'=(x',y',z')$ respectively each with a charge $-1$. Assume we are working on Hartree atomic units ...
6
votes
1answer
61 views
Inequality $(\int_0^1 |f|)^2 \leq \frac{1}{12}\int_0^1 {f'}^2$
Assume that $f\colon[0,1]\to\mathbb{R}$ is a differentiable function with $f'$ square integrable (if that's too weak or unpractical, assume that $f$ is continuously differentiable) such that $f(0)=f(1)...
1
vote
6answers
32 views
Finding the area bounded by $y = 2 {x} - {x}^2 $ and straight line $ y = - {x}$
$$
y =\ 2\ {x} - {x}^2
$$
$$
y =\ -{x}
$$
According to me , the area
$$
\int_{0}^{2}{2x\ -\ { x} ^2}\, dx \ + \int_{2}^{3}{\ {x} ^2\ -\ 2{x} }\, dx \\
$$
Which gives the area $ \frac{8}{3}$
But ...
1
vote
1answer
20 views
Volume of $x^2+y^2\leq4,\quad z=2+x^2+y^2,\quad z\geq-1$
Find the volume of the solid defined by $$x^2+y^2\leq4,\quad z=2+x^2+y^2,\quad z\geq-1.$$
I found the intersection of surfaces:
$$S\equiv\begin{cases}
x^2+y^2&=4\\
2+x^2+y^2&=z\\
z&=-1
\...
0
votes
3answers
55 views
Finding the integral of $\int_{0}^{2 \pi} \sin^n(x)$
$$\int_{0}^{2 \pi} \sin^n(x) = \, ?$$
The key step is to consider the complex integration $\int(z-\frac{1}{z})^n\frac{dz}{z}$ around the unit disk. Notice that $$(z-\frac{1}{z})^n\frac{1}{z} = \frac{...
0
votes
0answers
37 views
Finding $\lim_{x\to\infty}\int_0^x\frac{|\sin t|}tdt-\frac2\pi \ln x$
I want to evaluate
$$\lim_{x\to\infty}\int_0^x\frac{|\sin t|}tdt-\frac2\pi \ln x.$$
Related question:Asymptotic approximation of $\int_1^x \frac{| \sin t|}{t}dt$ as $x \to \infty$
It shows the $\...
5
votes
2answers
144 views
Ask a about hard integral of $\int_{0}^{\infty} \log x \log (\frac{a^2}{x^2}+1) \log(\frac{b^2}{x^2}+1)dx$
I want to evaluate the integral:
$$I(a,b)=\int_{0}^{\infty} \log x \log (\frac{a^2}{x^2}+1) \log(\frac{b^2}{x^2}+1)dx$$
Attempt:$$\frac{\partial ^2I}{\partial a\partial b}=4ab\int_{0}^{\infty}\frac{\...
1
vote
1answer
31 views
Integration of Legendre Polynomials with Dirac Delta function
The integral I need to solve is:
$$\int^{\pi}_{0} P_{l}(\cos(\theta))\delta(\theta-\frac{\pi}{2})\sin(\theta)d\theta.$$
where $\theta$ is the angle between $z$-axis and $x-y$ plane. Since I don't ...
2
votes
3answers
47 views
Exponential integral with square root in power
I have been trying to solve an integral. I know that the solution exists for the form
$$1- \dfrac{2}{\mathcal{R}^2}\int_{0}^{\mathcal{R}_{\mathcal{G}}} \exp (-\Phi r^{\alpha}) r\, {\rm d}r,$$
where ...
0
votes
1answer
40 views
Does this vector span $\mathbb{R}^{k}$?
Let $p_k(t)$ be a set of functions linearly independet of eachother, with $k>0$. Let $u$ be any function. Both p, and u are continuous and integrable. Let
$\alpha_k=\int_{0}^{T} p_k(T-\tau)u(\...
3
votes
3answers
228 views
Calculating $\int_0^{\pi/2} \frac{\ln(\sin x+\cos x)}{\sin x\cos x}\, dx$ [on hold]
How would you evaluate the following integral:
$$\int_0^{\pi/2} \frac{\ln(\sin x+\cos x)}{\sin x\cos x}\, dx$$
I can't see any helpful by parts or substitution ideas, so any help would be much ...
0
votes
1answer
64 views
Evaluating the integral $\int_0^π\frac{\sec^2x}{a^2+b^2\tan^{2}x}\, dx$
What will be the value of the following integral? I am getting the answer of $0$ but it's wrong.
$$\int_0^π\dfrac{\sec^2x}{a^2+b^2\tan^{2}x} \, dx$$
0
votes
1answer
79 views
Finding value of $\int^{2}_{0}\frac{\ln(1+x)}{1+x^2}dx$
Finding value of $\displaystyle \int^{2}_{0}\frac{\ln(1+x)}{1+x^2}dx$
Try: Let $\displaystyle I(a) = \int^{2}_{0}\frac{\ln(1+ax)}{1+x^2}dx$
Then $\displaystyle I'(a) = \int^{2}_{0}\frac{1}{(1+ax)(1+...
1
vote
0answers
23 views
High dimensional integral of exponentials
I am attempting to marginalize a probability density function. But I got stuck on the following integral
$$
\int_{-\infty}^\infty\cdots\int_{-\infty}^\infty
\frac{\exp(\pmb x A\pmb z)}
{|\exp(A\pmb z )...
2
votes
0answers
34 views
Integrating the product of a sine with two Bessel functions
I have a difficult integral again, which I would really prefer to be able to express analytically:
$$Y_{nlL}=\int_0^1 \sin \pi n r J_0( \gamma_l r )J_0(\gamma_L r) dr$$
Here $n=1,2,3,4,\dots$ and $\...
1
vote
0answers
48 views
Integration for Force of Lever
Archimedes' original computation of the area below a parabolic arc relied on a clever argument based on the law of the lever from physics. The law of the lever states that the force exerted on a lever ...
2
votes
2answers
27 views
Find the area bounded by $x=-y^2$ and $y=x+2$.
Question
Find the area bounded by $x=-y^2$ and $y=x+2$.
My Attempt
I know it is a very simple question to ask on MSE, but I don't know why I get stuck.
If you trace the graph, then the point of ...
1
vote
1answer
57 views
Approximation of an integral involving x and 1-x
I am looking for an approximation of the integral
$F(k,R)=\displaystyle\int_0^1\frac{\mathrm{d}x}{(Rx)^{-2}+(1-x)^{-k}}$,
that is valid to within 1% over the range $2<k<10$ and $R>1$. Is ...
15
votes
2answers
423 views
Is this Riemann sum formula for definite integral using of prime numbers true?
While answering another question in MSE, I had used the following result which I thought was a trivial consequence of the prime number theorem and equidistribution. However, I realized from the ...
-1
votes
0answers
68 views
Integration of $\int_{0}^{a} \frac{x^m}{(1+x)^n} dx$ [on hold]
I'd like to derive the integration:
$$I=\int_{0}^{a} \frac{x^m}{(1+x)^n}dx,(1)$$
where $a$ and $m$ are constant integers, $n\in \{0,1,2,\cdots\}$.
Could it be derived inductively? Thanks very much ...
4
votes
1answer
92 views
Evaluating complex integral $\int_{0}^{\pi} \frac {x \sin x}{1+a^2-2a(\cos x)} $ via different contour
I got an complex integral $\int_{0}^{\pi} \frac {x \sin
> x}{1+a^2-2a(\cos x)} $ for $a \ge 1$ and my given contour is a
rectangle such that $|Re(z)|\le \pi$ and $0 \le |Im(z)| \le h \to
\infty$....
5
votes
8answers
120 views
How to evaluate $\int_0^a\frac{x^4}{(a^2+x^2)^4}dx$?
I have to evaluate
$$\int_0^a\frac{x^4}{(a^2+x^2)^4}\,{\rm d}x$$
I tried to substitute $x=a\tan\theta$ which then simplifies to
$$\frac1{a^5}\int_0^\frac\pi4\sin^4\theta\cos^2\theta\, {\rm d}\...
2
votes
4answers
99 views
definite integral $\int_{0}^{\frac{\pi}{4}} \frac{\sin^2x\cos^2x}{\sin^3x+\cos^3x}dx$
$$\int_{0}^{\frac{\pi}{4}} \frac{\sin^2x\cos^2x}{\sin^3x+\cos^3x}dx$$
I tried $2$ or $3$ trigonometric transformations but it did not work. One of them is as follows
$$\frac{\sin^2x\cos^2x}{(\sin x+\...
2
votes
0answers
53 views
Exponential integral with power in square root
I have been trying to solve an integral. I know that the solution exists for the form
$$1- \dfrac{2}{\mathcal{R}^2_{\mathcal{G}}}\int_{0}^{\mathcal{R}_{\mathcal{G}}} \exp (-\Phi r^{\alpha}) r\, {\rm ...
4
votes
0answers
47 views
Integral of a triple product of Laguerre polynomials
I would like to know if there's an exact expression for this integral in terms of known elementary or special functions:
$$\int_0^\infty \exp \left(-\frac{a+b+c}{2}x \right) L_j (a x) L_k (b x) ...
4
votes
1answer
64 views
Definite integral on fractional part
Prove following definite integral:
$$\int_0^1\Bigl\{\frac{1}{x}\Bigr\}\ln(x)\,dx = \gamma_0+\gamma_1-1$$
Found in "Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis" ...
5
votes
2answers
60 views
How to integrate $\int_{1}^{e} (x+1)e^{x}\ln{x}dx $?
How to integrate $$ \int_{1}^{e} (x+1)e^{x}\ln{x}dx$$
I used following ways:
integration by parts
I first split the function into $$ \int_{1}^{e} (x)e^{x}\ln{x}dx + \int_{1}^{e} e^{x}\ln{x}dx$$
...
4
votes
1answer
43 views
Find Value of $g'(0)$ if $g(x)$ is inverse of $f(x)$, where $f(x)=\int_{2}^{x}\frac{1}{\sqrt{(1+t^4)}}dt$.
Find Value of $g'(0)$ if $g(x)$ is inverse of $f(x)$ where $$f(x)=\int_{2}^{x}\frac{1}{\sqrt{(1+t^4)}}dt.$$
I had tried following things
finding $f(x)$ by integration but failed
is $g(x)$ is ...
1
vote
1answer
56 views
integral of definite Meijer's G-function [closed]
I am working on my research and i have one integral seems difficult to me given as:
$$\int_0^\infty x^{-\omega}\exp(-\theta x)\large{G}_{1,2}^{1,1} \left( \beta x^{\alpha/2} \left|
\begin{array}{cc} ...
0
votes
1answer
28 views
Domain of integral of complex function from 0 to 1
If we have $t\in[0,1]\mapsto h(t)\in\mathbb{C}$ a continuous function and $H(z)=\int_0^1 \dfrac{h(t)}{t-z} dt$, is $\mathbb{C}\setminus[0,1]$ the domain of $H(z)$?
I was using the fact that ...
2
votes
4answers
41 views
Taking the derivative of an integral with x in the function and in the bound
I know how you would do this:
$${d\over dx}\int_2^{x^4} \tan (t^2)\,dt,$$ but how would you do this: $${d\over dx}\int_2^{x^4} \tan (x^2)\,dx.$$ I am confused on the argument $x$ being in the ...
-1
votes
4answers
101 views
Is $\int_a^b f(x)dx = \int_{-b}^{-a}f(x)dx$? [on hold]
There is step in my book in the solution of a question that I can't understand. It says :
$$\int_{3\sqrt3}^1\frac{dx}{1+x^2} + \int_{-1}^{-3\sqrt3}\frac{dx}{1+x^2}=2\int_{3\sqrt3}^1\frac{dx}{1+x^2}$$
...
0
votes
2answers
26 views
Right contour for integrating goniometric function with the $x^n$ as an argument
How would you integrate:
$\int_0^\infty \sin (x^n) \,dx$ $\;$for $n \gt 1$
I mean the result is via gamma function and there exists a formula for that gamma function but I struggled with the rooting ...
6
votes
3answers
146 views
Definite integral of $x\sin^n x$ from $0$ to $\pi/2$
How to find
\begin{equation*}
\int_0^{\pi/2} x\sin^n x dx
\end{equation*}
where $n$ is a positive integer? I tried $y=x-\pi/4$ and that gives
\begin{equation*}
\frac{1}{2^{n/2}}\frac{\pi}{4}\int_{-\pi/...
2
votes
1answer
88 views
Integral $\int_0^1 \frac{x\ln\left(\frac{1+x}{1-x}\right)}{\left(\pi^2+\ln^2\left(\frac{1+x}{1-x}\right)\right)^2}dx$
The goal is to show (preferably without contour integration, as my knowledge is pretty limited there, but if you can do it that way there is no problem to share it) that: $$I= \int_0^1 \frac{x\ln\left(...
6
votes
2answers
127 views
Calculate$\int\limits_{-2}^{0} \frac{x}{\sqrt{e^x+(x+2)^2}}dx$ [duplicate]
Calculate $$\int\limits_{-2}^{0} \frac{x}{\sqrt{e^x+(x+2)^2}}dx$$
First I tried the substitution $t=x+2$ and obtained $$\int\limits_{0}^{2} \frac{t-2}{\sqrt{e^{t-2}+t^2}}dt$$ and than I thought to ...
6
votes
0answers
217 views
An $\operatorname{erfi}(x)e^{-x^2}$ integral
I want to find an elementary evaluation of
$$I=\int_0^\infty \left(\frac{\sqrt\pi}2\operatorname{erfi}(x)e^{-x^2}-\frac1{1+2x}\right)dx$$
where $\operatorname{erfi}(x)=\frac{2}{\sqrt\pi}\int_0^...
0
votes
1answer
47 views
exponential integral with arbitrary power of variable
I have been trying to solve a simple mathematical integral. I know that the solution exists but it is not being verified by Matlab. Here is the integral that I am solving along with its answer which I ...
7
votes
3answers
132 views
A double sum or a definite integral.
I am trying to evaluate the following double sum
\begin{eqnarray*}
\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{(-1)^{n+m}}{n(3n+m)}.
\end{eqnarray*}
Using the integral trick
\begin{eqnarray*}
\...
2
votes
3answers
72 views
How to tell if $\sum_{n=1}^\infty\frac{\ln(n)}{n^2}$ converges using Integral Test?
So I have this problem:
Determine whether or not the following infinite series converge or diverge. State what test you used.
$$\sum_{n=1}^\infty\frac{\ln(n)}{n^2}$$
So I decided to do the ...
0
votes
1answer
21 views
How to solve this problem about leveling a ground?
We are given this plot
from which I extract that $f(x)=\frac{x^2}{1600}-\frac{x}{4}+125$. I am asked to find the height at which the ground should be leveled, in such a way that that the exceeding ...
1
vote
4answers
64 views
Evaluate $\lim\limits_{x\to 0}\frac{\int\limits_0^{x^2}\sec^2 t \, dt}{x\sin x}$
Question :
Evaluate $\lim\limits_{x\to 0}\dfrac{\int\limits_0^{x^2}\sec^2 t \, dt}{x\sin x}$.
I hahe not got any idea how to evaluate. How can I do this?
0
votes
1answer
93 views
What is $\int\limits_{-\infty}^{\infty} \frac{\cos(x)}{x} dx$
$$\int\limits_{-\infty}^{\infty} \frac{\cos(x)}{x} dx$$
I tried thinking along the following lines, but I get conflicting answers...
$\frac{\cos(x)}{x}$ is an odd function. So the integral must be $...
3
votes
2answers
98 views
Evaluating $\int_0^\infty\operatorname{erfi}(x)e^{-x^2}\frac{dx}x$
I want to evaluate
$$I=\int_0^\infty\operatorname{erfi}(x)e^{-x^2}\frac{dx}x$$
where $\operatorname{erfi}(x)=\frac{2}{\sqrt\pi}\int_0^x e^{t^2}dt$.
I can only prove this integral converges.
Not-...
1
vote
2answers
64 views
Find $\lim_{r \to \infty} \int_{0}^{\pi} e^{-r\cos^2(\theta)} d\theta$
$\displaystyle \lim_{r \to \infty} \int_{0}^{\pi} e^{-r\cos^2(\theta)} d\theta$
Here is what I did:
Since $e^{-r\cos^2(\theta)} $ is continuous on $[0,\pi]$ for any fixed $r$, we can use MVT:
There ...
0
votes
0answers
29 views
Solving definite integrals with (ax+b)^n
I'm trying to solve the following problem without the chain rule or other "method", only formulas:
$$
g(x)=\int _{3x}^{x^3}\left(t^3+1\right)^{10}dt
$$
I'm used this formula, which is very helpful:
...
0
votes
0answers
25 views
asymptotic expansion of integral of Bessel function?
The integral is
$\int_0^\infty dx\frac{x J_1(\sqrt(y)x/\pi)^2}{y\sinh^2(x)}$
I want to compute it in large $y$ expansion. I have checked numerically that the integral is convergent for all positive ...
3
votes
1answer
96 views
Why isn't $\int\limits_0^{\pi/2}\tan(2x)\ dx$ equal to zero?
I know about the improper integral issue, but analyzing the graph we see that $\tan (x)$ is odd around $\pi/4$, so if we say that the positive area is $+\infty$, we know that the other one is $-\infty$...
0
votes
1answer
37 views
Gamma function: relating infinite product and integral definitions. Stuck on a step in the proof.
This has been asked and answered before, but at a level far above anything I could understand.
When proving that: $\Gamma(z+1)=\int_0^\infty x^{z}e^{-x} dx$
is the same as: $\Gamma(z+1)=N^z \frac{1}...
4
votes
2answers
134 views
How to prove the integral $\int_{0}^{\infty} x^{-1}\sin (x+x^{-1})\,dx=\pi J_0(2)$
$$I=\int_{0}^{\infty} \frac{\sin \left(x+\frac{1}{x}\right)}{x}dx=\pi J_0(2)$$
I'v found:$$I=2\int_{0}^{\infty}\frac{\sin \left(x^2+\frac{1}{x^2}\right)}{x}dx=3\int_{0}^{\infty}\frac{\sin \left(x^3+\...
