All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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33
votes
5answers
791 views
+200

How to find ${\large\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$

Please help me to find a closed form for this integral: $$I=\int_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx\tag1$$ I suspect it might exist because there are similar integrals having closed forms: ...
0
votes
0answers
5 views

Trigonometric integral: $\int_{25\pi/4}^{53\pi/4} \frac{1}{(1+2^{\sin x})(1+2^{\cos x})}\,dx$

Is it possible to evaluate the following in a closed form? $$\int_{25\pi/4}^{53\pi/4} \frac{1}{(1+2^{\sin x})(1+2^{\cos x})}\,dx$$ I found the above definite integral at I&S but the solution is ...
0
votes
0answers
5 views

Laplace transform convolution

I can't seem to get this Laplace working using the convolution method. $H(s) = \frac{1}{s^2(s+2)}$ Which I can't get to work using convolution. So I am separating it into $\frac{1}{s^2} * ...
2
votes
1answer
37 views

How do I Solve This Kind of Differential Equation?

How do I solve this differential equation? $$y(2x+y^2)dx+x(y^2-x)dy=0$$
1
vote
1answer
30 views

Calc II - Definite integral of sqrt(t^2 + t) from 2x to 1?

How do I find $$\int_1^{2x}\sqrt{t^2 + t}$$ with only knowledge from a Calculus I course? I've tried plugging this puppy into Wolfram Alpha and other integral solvers, which report it as solvable ...
0
votes
1answer
23 views

When can we exchange sum limit and integral

A simple question: when can wo exchange sum and integral? $$\sum_{n=0}^\infty\int f_n(x)dx=\int\sum_{n=0}^\infty f_n(x)dx=\int f(x)dx$$ $$\lim_{n\rightarrow \infty}\int f_n(x)dx=\int\lim_{n\rightarrow ...
1
vote
2answers
95 views

How to prove that $\int_{-1}^{1}\exp\left(\frac{1}{x^2-1}\right) \ dx=1$?

I have some trouble to prove that $$\int_{-1}^{1}\exp\left(\frac{1}{x^2-1}\right) \ dx=1\ ? $$
1
vote
1answer
68 views

How can calculate this integral?

I have the following integral as: $$\int_{0}^{\infty}x\exp\left(-\frac{x^2+a^2}{2}\right)I_0(ax)dx.$$ where $I_0(.)$ is Bessel function of zero degree. Can any one help me calculataing this integral? ...
0
votes
1answer
16 views

Laplace transform on a non-standard sort of problem

I don't know where a laplace comes into play here: $\ddot{a}+2a=0,a(0)=b_1,\dot{a}(0)=b_2$ I am meant to solve the above using a Laplace transform, but I don't see how I would use it here? I ...
2
votes
2answers
31 views

How to simplify the integral of $\int\frac{\cos(8x)}{\cos(4x)+\sin(4x)}dx$?

So I am trying to integrate this problem $\int\frac{\cos(8x)}{\cos(4x)+\sin(4x)}dx$, and my professor went over it in class and went from $\int\frac{\cos(8x)}{\cos(4x)+\sin(4x)}dx \rightarrow ...
3
votes
3answers
49 views

Integral of $\cos(\cos x)$ over $[0,2\pi]$

How to compute the following integral? $$\mathcal{J}_2=\int_{0}^{2\pi}\cos(\cos t)\,dt$$ I'm trying to compute this integral, but I have no idea of how to do it, can someone help me?
7
votes
3answers
161 views

Hard Definite integral involving the Zeta function

Prove that: $$\displaystyle \int_{0}^{1}\frac{1-x}{1-x^{6}}{\ln^4{x}} \ {dx} = \frac{16{{\pi}^{5}}}{243\sqrt[]{{3}}}+\frac{605\zeta(5)}{54} $$ I was able to simplify it a bit by substituting ${y = ...
2
votes
3answers
118 views

Limit of a Riemann Sum and Integral

I've been trying to solve this problem, but I haven't been able to calculate the exact limit, I've just been able to find some boundaries. I hope you guys can help me with it. Let $f:[0,1] \to ...
4
votes
1answer
89 views

Calculation of integral with Bessel function

I have a trouble with to calculating (or bounding from above) the following integral: $$ \int_{-\infty}^{\infty}\left(\frac{J_2(x)}{x^2}\right)^p\, dx, \quad p\geq 1, $$ where $J_2(x)$ is a Bessel ...
3
votes
3answers
75 views

Changing order of integration (multiple integral)

Prove $$ \int_0^a\left( \int_0^x \left( \int_0^y \left( \int_0^z f(u) \, du \right) dz \right) dy \right) dx = \int_0^a \frac {(a-t)^3}{3!} f(t) dt $$ where $a$ is constant. So I began with two ...
6
votes
2answers
57 views

Trigonometric functions expressed as definite integrals with Bessel functions

Prove that $$\frac{\sin(x)}{x}=\int_0^\frac{\pi}{2}J_0(x\cos(\theta))\cos(\theta)\,d\theta \tag{a}$$ $$\frac{1-\cos(x)}{x}=\int_0^\frac{\pi}{2}J_1(x\cos(\theta))\,d\theta \tag{b}$$ Hint: ...
4
votes
1answer
186 views

Integral with Bessel function

Let $n$ be half an odd integer, say $n=k+1/2, k \in \mathbb{N}$. Let $q\geq 1$. I would like to calculate (or approximate) the following integral: $$ \int_0^{\infty}\left(\sqrt{\frac{\pi}{2}}\cdot ...
8
votes
2answers
191 views

A closed form for $\int_{0}^{\pi/2}\frac{\ln\cos x}{x}\mathrm{d}x$?

The following integrals are classic, initiated by L. Euler. \begin{align} \displaystyle \int_{0}^{\pi/2} x^3 \ln\cos x\:\mathrm{d}x & = -\frac{\pi^4}{64} \ln 2-\frac{3\pi^2}{16} ...
12
votes
2answers
179 views

Integral $\int_0^{\Large\frac{\pi}{4}}\left(\frac{1}{\log(\tan(x))}+\frac{1}{1-\tan(x)}\right)dx$

I am wondering if anyone would know how to evaluate this integral: $$\int_{0}^{\Large\frac{\pi}{4}}\left(\frac{1}{\log(\tan(x))}+\frac{1}{1-\tan(x)}\right)dx.$$ I've tried, unsuccessfully, the change ...
4
votes
1answer
71 views

A closed-form of $\frac{1}{2}\int_0^\infty\left[\frac{x^2\cos x}{\cosh 2x-\cos x}-\frac{2x^2}{e^{4x}-2e^{2x}\cos x+1}\right]\,dx$

I am looking for a closed-form of this integral \begin{equation} \frac{1}{2}\int_0^\infty\left[\frac{x^2\cos x}{\cosh 2x-\cos x}-\frac{2x^2}{e^{4x}-2e^{2x}\cos x+1}\right]\,dx \end{equation} I ...
1
vote
1answer
48 views

Find the coefficients in quadrature formula on $[0,1]$ with the nodes at $1/4$, $1/2$, $3/4$

In my worksheet I was given a question about numerical integration that says: Find the formula for $\int_{0}^{1}f(x)dx=A_{0}f(\frac{1}{4})+A_1f(\frac{1}{2})+A_2f(\frac{3}{4})$ I suppose the goal ...
1
vote
0answers
27 views

Help with functional integral

I'm stuck on how to do a functional integral. The integration I'm trying to do is of this form $\frac{\partial}{\partial B(\tau)} \left[ \exp\left(-B^2(\tau)\right)+\int_{0}^{\tau} ...
0
votes
0answers
12 views

Integrating differential forms over a box

I've only ever seen examples of integrating a differential form over a curve C involving defining a parameterization. I have seen people integrate 1 forms over a box without defining a ...
-1
votes
1answer
54 views

Natural Cubic Spline Confusion

Find the natural cubic spline which interpolates the data points $(1,0),\; (2,1),\; (3,0), \; (4,1), \; (5,0) $. I know how to check if a piecewise function is a natural cubic spline, but I don't ...
1
vote
0answers
34 views

How can I calculate the force that is applied on a tube by an another tube?

Let's say there is two tubes (cylinders with no tops or bottoms) with charges $q_1$ and $q_2$, radii $b_1$ and $b_2$, lengths $\ell_1$ and $\ell_2$. These tubes are located along the axis of each ...
4
votes
0answers
66 views

is there closed form for $\int_0^{\pi/4}\exp(-\sum_{n=1}^{\infty}\frac{\tan^{2n}x}{n+a})dx$

Is there closed form for $$I(a)=\int_0^{\pi/4}\exp(-\sum_{n=1}^{\infty}\frac{\tan^{2n}x}{n+a})dx $$where is $a\in (-1,3)$ I've tried with $\tan x=u$ and I got the result of sum in term of ...
6
votes
0answers
90 views

Quaternion integration

If the angular velocity is changing continuously, the following holds true $ q(t)=q(0)\exp\left({\int_{0}^{t}\frac{q_\omega(\tau)}{2}\ d\tau}\right) \tag 1$ Specifications and Data $q(t),q(0)$ ...
1
vote
0answers
18 views

Will Gauss quadrature numerical integration work with a variable dx

The question kind of says it all, but I'm reading about Gauss quadrature from here: http://www.damtp.cam.ac.uk/lab/people/sd/lectures/nummeth98/integration.htm which gives an equation of this form: ...
2
votes
0answers
9 views

Calculating $\text{D}g$ of $g(x,y) = \int_\frac1x^1\frac1t\exp(t^3x^2y)\text{d}t$

Let $g:(1,\infty)^2\to\mathbb{R}$ be given by $$g(x,y) = \int_\frac1x^1\frac1t\exp(t^3x^2y)\text{d}t.$$ How can I calculate $\text{D}g$ using parameter-dependent integrals?
2
votes
0answers
33 views

What is an example of a function that is measurable but not strongly measurable?

Let $(\Omega, \Sigma)$ be a measurable space and $X$ a Banach space. Let $f: \Omega \rightarrow X$. $f$ is called measurable if every the preimage of every Borel set in $X$ is an element of ...
11
votes
3answers
331 views

Another integral for $\pi$

Here is a new integral for $\pi$. $$\int_{0}^{1}\sqrt{\frac{\left\{1/x\right\}}{1-\left\{1/x\right\}}}\, \frac{\mathrm{d}x}{1-x} = \pi $$ where $\left\{x\right\}$ denotes the fractional part of ...
6
votes
2answers
108 views

Integral $ \int_{0}^1 \sqrt{\frac{\ln{x}}{x^2-1}} dx$

Please help evaluating this integral $$ \large\int_{0}^1 \sqrt{\frac{\ln{x}}{x^2-1}} dx$$ Mathematica could not evaluate it in a closed form. Numerically it is about ...
6
votes
3answers
136 views

Find $\int_0^\pi \sin(x)\,dx$ explicitly

A book asks me to prove that: $$\int_0^{\pi}\sin(x)\,dx = 2$$ Using the identity: $$\sin\left(\frac{\pi}{n}\right) + \sin\left(\frac{2\pi}{n}\right) + \cdots + \sin\left(\frac{n\pi}{n}\right) = ...
20
votes
4answers
971 views

How to find $\int x^{1/x}\mathrm dx$

EDIT: The full answer has been posted by myself. Feel free to check the logic within. How does one indefinitely integrate a function in the form of $$f(x)=x^{1/x}$$ Looking at all the things that I ...
2
votes
0answers
79 views
+50

Verifying an antiderivative found in any integral table

If $a > 0$, and $0 < b < c$. \begin{equation*} \int \frac{1}{b + c\sin(ax)} \, {\mathit dx} = \frac{-1}{a\sqrt{c^{2} - b^{2}}} \, \ln\left\vert\frac{c + b\sin(ax) + \sqrt{c^{2} - ...
21
votes
3answers
535 views
+500

Integral ${\large\int}_0^\infty\frac{\ln x}{1+x}\sqrt{\frac{x+\sqrt{1+x^2}}{1+x^2}}\ \mathrm dx$

Please help me to evaluate this integral: $$ I={\large\int}_{0}^{\infty}{\ln\left(x\right) \over 1 + x}\, \,\sqrt{\,x + \sqrt{\,1 + x^{2}\,}\, \over 1 + x^{2}\,}\,\,{\rm d}x.\tag1 $$ Mathematica could ...
4
votes
1answer
145 views
+50

Area of ​​the intersection of two discs: Integral solution?

Here is the problem : We consider two circles that intersect in exactly two points. There $O_1$ the center of the first and $r_1$ its radius. There $O_2$ the center of the second and $r_2$ its ...
2
votes
3answers
101 views

What is $\operatorname{Ei}(x)$?

I was trying to solve $$\int\frac{e^x - e^{-x}}{x}\,dx$$ But I have no idea how to do it and the calculator said to use a common integral that I don't know what it means.
28
votes
9answers
2k views

Closed form for $\int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$

I've been looking at $$\int\limits_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$$ It seems that it always evaluates in terms of $\sin X$ and $\pi$, where $X$ is to be determined. For example: ...
31
votes
3answers
536 views

How to evaluate $\int_0^\infty\operatorname{erfc}^n x\ \mathrm dx$?

I successfully evaluated these integrals: $$\int_0^\infty\operatorname{erfc}x\ \mathrm dx=\frac1{\sqrt\pi},\tag1$$ $$\int_0^\infty\operatorname{erfc}^2x\ \mathrm dx=\frac{2-\sqrt2}{\sqrt\pi}\tag2,$$ ...
33
votes
9answers
5k views

Proof of $\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.$

I am looking for a short proof that $$\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.$$ What do you think? It is kind of amazing that $$\int_0^\infty \frac{\sin x}{x} \mathrm ...
34
votes
8answers
1k views

How to evaluate $I=\displaystyle\int_0^{\pi/2}x^2\ln(\sin x)\ln(\cos x)\ \mathrm dx$

Find the value of $I=\displaystyle\int_0^{\pi/2}x^2\ln(\sin x)\ln(\cos x)\ \mathrm dx$ We have the information that $J=\displaystyle\int_0^{\pi/2}x\ln(\sin x)\ln(\cos x)\ \mathrm ...
34
votes
1answer
647 views

Integral $\int_0^1\frac{x^9\left(x^4+x^2-x-1-5\ln x\right)}{\left(x^{10}-1\right)\ln x}\mathrm dx$

A friend of mine sent me an integral that she had not been able to crack, and me neither. It comes with a result, but without a proof (I suppose it originated in some math contest). Could you please ...
3
votes
1answer
98 views

When may we ignore the limits of integration?

When we try to evaluate an integral such as, say $$\int_a^b{f(x)dx}$$ there is often the case that we can analytically find $$\int{f(x)dx}$$ a little faster (imagine leaving away the evaluation ...
34
votes
4answers
1k views

Integral $\int_0^1\frac{\ln\left(x+\sqrt2\right)}{\sqrt{2-x}\,\sqrt{1-x}\,\sqrt{\vphantom{1}x}}\mathrm dx$

Is there a closed form for the integral $$\int_0^1\frac{\ln\left(x+\sqrt2\right)}{\sqrt{2-x}\,\sqrt{1-x}\,\sqrt{\vphantom{1}x}}\mathrm dx.$$ I do not have a strong reason to be sure it exists, but I ...
36
votes
1answer
981 views

To evaluate $\int_0^{+\infty} \frac{\;\mathrm dx}{\sqrt[3]{x^3+a^3}\sqrt[3]{x^3+b^3}\sqrt[3]{x^3+c^3}}$

$$f(a,b)=\int_0^{+\infty} \frac{\;\mathrm dx}{\sqrt{x^2+a^2}\sqrt{x^2+b^2}}$$ To use Landen's transformation $$f(a,b)=\int_0^{+\infty} \frac{\;\mathrm ...
38
votes
5answers
4k views
38
votes
4answers
1k views

An integral involving Airy functions $\int_0^\infty\frac{x^p}{\operatorname{Ai}^2 x + \operatorname{Bi}^2 x}\mathrm dx$

I need your help with this integral: $$\mathcal{K}(p)=\int_0^\infty\frac{x^p}{\operatorname{Ai}^2 x + \operatorname{Bi}^2 x}\mathrm dx,$$ where $\operatorname{Ai}$, $\operatorname{Bi}$ are Airy ...
41
votes
1answer
1k views

Conjecture $\int_0^1\frac{\mathrm dx}{\sqrt{1-x}\ \sqrt[4]x\ \sqrt[4]{2-x\,\sqrt3}}\stackrel?=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi$

$$\int_0^1\frac{\mathrm dx}{\sqrt{1-x}\ \sqrt[4]x\ \sqrt[4]{2-x\,\sqrt3}}\stackrel?=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi\tag1$$ The equality numerically holds up to at least $10^4$ decimal digits. ...