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Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

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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
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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)...
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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 ...
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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 \...
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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{...
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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 $\...
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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{\...
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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 ...
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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 ...
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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(\...
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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 ...
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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. $$\int_0^π\dfrac{\sec^2x}{a^2+b^2\tan^{2}x} \, dx$$
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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+...
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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 )...
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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 $\...
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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 ...
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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 ...
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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 ...
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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 ...
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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
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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 $0 \le |Im(z)| \le h \to \infty$....
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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
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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
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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
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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
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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" ...
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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
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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 ...
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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} ...
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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 ...
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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 ...
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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 : $$\int_{3\sqrt3}^1\frac{dx}{1+x^2} + \int_{-1}^{-3\sqrt3}\frac{dx}{1+x^2}=2\int_{3\sqrt3}^1\frac{dx}{1+x^2}$$ ...
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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 ...
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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/...
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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
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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
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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^...
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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
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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
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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 ...
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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 $-\infty$...
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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+\...