0
votes
3answers
40 views

Using the comparison test to evaluate $\int_1^\infty\frac{1}{1+x^2+16x^4}dx$?

So using the comparison test to evaluate $\int_1^\infty\frac{1}{1+x^2+16x^4}dx$, and we're given $\int_1^\infty\frac{1}{4x^2}dx$. So I have been trying to set up an inequality to use, but I can't seem ...
1
vote
4answers
82 views

Does the following integral converge: $\int_6^{\infty}\frac{dx}{\sqrt{1+x^2}}$

Does the following integral converge: $$\int_6^{\infty}\frac{dx}{\sqrt{1+x^2}}$$ I suppose we have to solve such problems by comparison test. All the integrals I tried so far do not fit the ...
1
vote
3answers
30 views

Finding the integral of $\int_{-\infty}^{\infty}e^{-|4x|}$.

So I am trying to find the integral of $\int_{-\infty}^{\infty}e^{-|4x|}$. I know the integral converges, and I know the answer as well, but I am confused on how to get the correct answer. My problem ...
17
votes
2answers
153 views
+300

Closed-forms for several tough integrals

These integrals came up in the process of finding solution to Vladimir Reshetnikov's problem. I wonder if there are closed-forms for the following integrals: \begin{array}{1,1} &[\text{1}] ...
5
votes
2answers
53 views

Integral with rational functions of powers and exponentials

Any ideas how to solve: \begin{equation} \int_0^\infty x^{n+\frac{1}{2}} (e^{a x }-1)^{-\frac{1}{2}} e^{i x t} dx \end{equation} where $a$ and $t$ are real, positive constants; $n$ is a positive ...
15
votes
3answers
244 views

Integral $\int_{0}^1\frac{\ln\frac{3+x}{3-x}}{\sqrt{x(1-x)}}dx$

I have a problem with the following integral: $$ \int_{0}^{1}\ln\left(\,3 + x \over 3 - x\,\right)\, {{\rm d}x \over \,\sqrt{\,x\left(\,1 - x\,\right)\,}\,} $$ The first idea was to use the ...
11
votes
2answers
231 views
+150

Integral of Combination Log and Inverse Trig Function

Does the following integral have a closed-form \begin{equation}\int_0^1\frac{\ln x}{1+x}\arccos(x)\,dx\,?\end{equation} This integral has been posted in Integral and Series a week ago but it ...
2
votes
4answers
107 views

Integral $\int_{1}^{\infty} \frac{\log^3 x}{x(x-1)} dx$

How do I arrive at the closed form expression of the integral $$\displaystyle\int_{1}^{\infty} \dfrac{\log^3 x}{x(x-1)}dx$$ Most probably the closed form is $\dfrac{\pi^4}{15}$
0
votes
3answers
43 views

General closed form of an integral

I once asked a question about how to integrate the reciprocal of the square root of cosine. Is there a general closed form for the integral $$\int_{0}^{\theta_o} \dfrac{1}{\sqrt{\cos \theta-\cos ...
6
votes
4answers
124 views

Calculus Question: Improper integral $\int_{0}^{\infty}\frac{\cos(2x+1)}{\sqrt[3]{x}}dx$

How to evaluate integral $$\int_{0}^{\infty}\frac{\cos(2x+1)}{\sqrt[3]{x}}dx?$$ I tried substitution $x=u^3$ and I got $3\displaystyle\int_{0}^{\infty}u \cos(2u^3+1)du$. After that I tried to use ...
2
votes
0answers
42 views

An integral with a decaying exponential with rational exponent

I was working on some mathematical derivations while I faced this integral: $$\Large \int_0^\infty x^{\alpha-1}e^{-\beta x} e^{-\lambda \left[\frac{x^2}{2x+\eta}\right]}\ \mathrm{d}x \quad .$$ Does ...
1
vote
3answers
40 views

Compute variance, using explicit PDF

I'm trying to get $\text{Var}(x)$ of $f(x) = 2(1+x)^{-3},\ x>0$. Please tell me if my working is correct and/or whether there is a better method I can use to get this more easily. $$ ...
0
votes
2answers
32 views

Help finding k. Issue with integration

Let the continuous random variable $X$ have a probability density function $f(x)$ such that $$f(x) = k(1+x)^{-3}, x>0$$ $=0$ elsewhere Find k This is what I tried: $\int_0^\infty k(1+x)^{-3}dx ...
2
votes
3answers
45 views

Improper integral of rational function $k^2/(1+a^2k^2)^2$

I've got the integral $\int^\infty_{-\infty} dk \frac{k^2}{(1+a^2 k^2)^2}$ where $a$ is a real number. I can't seem to find a $u$-substitution or trigonometric substitution that will work. Any ...
0
votes
1answer
24 views

Solving Coulomb Integral in 1D

I am trying to solve the following Coulomb integral of two gaussians: $$ \int_{- \infty}^{ \infty}dx1\int_{- \infty}^{ \infty} \frac{e^{-b1 (x1-c1)^2}e^{-b2 (x2-c2)^2}}{\left | x1-x2 \right |}dx2, $$ ...
2
votes
5answers
108 views

How to prove this integral?

I was confused about an integral showing on my teacher's slide, could anyone tell me how is the following integral derived? $$ \int^\infty_{-\infty} x^{2k} e^{-\frac{x^2}{2\sigma^2}} \; \mathrm{d}x ...
5
votes
2answers
306 views

A logarithmic integral

How can I evaluate following logarithmic integral: $$\int\limits_0^1 \frac{\ln x\ln ( 1 - zx )}{1 - x} dx$$
8
votes
7answers
208 views

Evaluate $ \int_{0}^{1} \ln(x)\ln(1-x)\,dx $

Evaluate the integral, $$ \int_{0}^{1} \ln(x)\ln(1-x)\,dx$$ I solved this problem, by writing power series and then calculating the series and found the answer to be $ 2 -\zeta(2) $, but I don't ...
9
votes
4answers
454 views

Looking for closed-forms of $\int_0^{\pi/4}\ln^2(\sin x)\,dx$ and $\int_0^{\pi/4}\ln^2(\cos x)\,dx$

A few days ago, I posted the following problems Prove that \begin{equation} \int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24}\\[20pt] -\int_0^{\pi/2}\ln^3(\cos ...
4
votes
4answers
165 views

Prove that $\int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24}$

Prove that \begin{equation} \int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24} \end{equation} I tried to use by parts method and ended with \begin{equation} \int \ln^2(\cos ...
17
votes
2answers
273 views

Prove $\displaystyle \int_{0}^{\pi/2} \ln \left(x^{2} + (\ln\cos x)^2 \right) \, dx=\pi\ln\ln2 $

How to prove\begin{equation} \int_{0}^{\pi/2} \ln \left(x^{2} + (\ln(\cos x))^2 \right) \, dx=\pi\ln\ln2 \end{equation} I don't know how to answer it. When I asked this integral to my brother, ...
6
votes
1answer
104 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 ...
42
votes
5answers
1k views

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: ...
3
votes
2answers
89 views

cosine integral

Show that $$\int_0^x \frac{1-\cos(t)}{t}=\gamma+\ln(x)-\operatorname{Ci}(x)$$ where $$\operatorname{Ci}(x)=-\int_x^\infty \frac{\cos(t)}{t} \, dt$$ and gamma is an euler-mascheroni constant. I did as ...
7
votes
2answers
192 views

improper integral containing $\sqrt{\cos x-\dfrac{1}{\sqrt 2}}$ in the denominator

How do i find the value of this integral-- $$I=\displaystyle\int_{0}^{\pi/4} \frac{\sec^2 x \ dx}{\sqrt {\cos x-\dfrac{1}{\sqrt 2}}}$$ I came across this integral too in physics.
1
vote
1answer
126 views

How to find this integral $\int_{0}^{\infty}\dfrac{f(x)}{g(x)}dx$ [duplicate]

show that: $$I=\int_{0}^{\infty}\dfrac{x^8-4x^6+9x^4-5x^2+1}{x^{12}-10x^{10}+37x^8-42x^6+26x^4-8x^2+1}dx=\dfrac{\pi}{2}$$ I found this : ...
1
vote
2answers
86 views

Solving integral that contain exponential function and lower incomplete gamma function

I have the following integral; $$y=\int_0^\infty\frac{e^{-xf}}{m+x}\gamma(a,hx)~dx$$ where $f,m,h\in\mathbb{R}^+$ , $a\in\mathbb{N}$ , $\gamma\left(a,h x\right)$ is the lower incomplete gamma ...
6
votes
0answers
78 views

Dealing with an integral: can we go any farther?

I meet an integral, but it is beyond my ability. $$ {\rm I}\left(a\right) = \int_{a}^{1}{\arcsin\left(\,\sqrt{\,{1 - x^{2} \over 1 - a^{2}}\,}\,\right) \over x + 1}\,{\rm d}x, 0\le a <1. $$ I can ...
4
votes
4answers
199 views

Definite integral $\int_{0}^{\infty}e^{-u}\frac{1}{\left(\sqrt{1+(h+u)^{2}}\right)^{5}}du$

Hi guys I have the following definite integral to solve: $$\int_{0}^{\infty}e^{-u}\frac{1}{\left(\sqrt{1+(h+u)^{2}}\right)^{5}}du$$ is it possible to obtain an analytic expression? And if not why? ...
2
votes
2answers
212 views

An Improper Integral

I need help with this integral: $\Large {\int_0^\infty \frac{dx}{x\sqrt{1+x}}} $ What I did: Substitute $\sqrt {1+x} = t$. Then the integral turns into $ \int_1^\infty 2dt/(t^2-1) $. Now I replaced ...
0
votes
1answer
133 views

Finding the value of $\int_{0}^{1} \frac{\sin^2 x}{x^2}dx$

I would like to find the exact value of $$\int_{0}^{1} \frac{\sin^2 x}{x^2}dx.$$ First of all we know that it exists and must be $\hspace{0.1cm}$$\leq1$$\hspace{0.1cm}$ because$\hspace{0.1cm}$ ...
16
votes
2answers
454 views

Closed form for the integral $\int_{0}^{\infty}\frac{\ln^{2}(x)\ln(1+x)}{(1-x)(x^{2}+1)}dx$

Here is a challenging one maybe some would like a go at. Show that: ...
7
votes
1answer
75 views

Closed form for integral of integer powers of Sinc function

(Edit: Thank you Vladimir for providing the references for the closed form value of the integrals. My revised question is then to how to derive this closed form.) For all $n\in\mathbb{N}^+$, ...
2
votes
3answers
76 views

Evaluate $\frac{1}{a}\int_0^\infty{x^2}e^{-\frac{x^2}{2a}}\,dx$

Evaluate the following integral: $$\frac{1}{a}\int_0^\infty{x^2}e^{-\large\frac{x^2}{2a}}\,dx.$$
12
votes
5answers
309 views

The other ways to calculate $\int_0^1\frac{\ln(1-x^2)}{x}dx$

Prove that $$\int_0^1\frac{\ln(1-x^2)}{x}dx=-\frac{\pi^2}{12}$$ without using series expansion. An easy way to calculate the above integral is using series expansion. Here is an example ...
9
votes
3answers
132 views

Proof of $\int_0^\infty \frac{x^{\alpha}dx}{1+2x\cos\beta +x^{2}}=\frac{\pi\sin (\alpha\beta)}{\sin (\alpha\pi)\sin \beta }$

I found a nice formula of the following integral here $$\int_0^\infty \frac{x^{\alpha}dx}{1+2x\cos\beta +x^{2}}=\frac{\pi\sin (\alpha\beta)}{\sin (\alpha\pi)\sin \beta }$$ It states there that ...
12
votes
3answers
221 views

Prove that $\int_0^1\frac{\ln(1-x)\ln^2x}{x-1}dx=\frac{\pi^4}{180}$

Prove that (please) $$\int_0^1\frac{\ln(1-x)\ln^2x}{x-1}dx=\frac{\pi^4}{180}$$ I've tried using Taylor series and I ended up with $$-\sum_{m=0}^\infty\sum_{n=1}^\infty\frac{2}{n(m+n+1)^3}$$ I am ...
2
votes
1answer
125 views

Show that $\int^{\infty}_0 \frac{\sin^4 x}{x^4}=\frac{\pi} 3$. [duplicate]

Show that $$\int^{\infty}_0 \left( \frac{\sin x}{x}\right)^4=\frac{\pi} 3$$ Although I know the integral with the index is $1$ and $2$, I have no idea on this one. Please help.
9
votes
3answers
288 views

A probabilistic integral $\int_{-\infty}^{\infty}e^{-x^2/2\sigma^2}\arcsin\left(1-2\left|\lfloor x\rceil-x\right|\right)\,dx$

In my probabilistic studies, a tough integral appeared. Note that $\lfloor x\rceil$ is not the floor function; it is the nearest integer function. Up to some constants, it appears in a Buffon-like ...
2
votes
3answers
98 views

Evaluate $\int_{1}^{\infty} \frac{\ln{(2x-1)}}{x^2} $

$$\int_{1}^{\infty} \frac{\ln{(2x-1)}}{x^2} dx$$ My approach is to calc $$\int_{1}^{X} \frac{\ln{(2x-1)}}{x^2} dx$$ and then take the limit for the answer when $X \rightarrow \infty$ However, I must ...
4
votes
3answers
285 views

Prove $\int_{\mathbb{R^{+}}} \frac{\sin^3 {(\pi x^2)} \cos {(4x^2)}}{x^5} dx=\frac{\pi}{32} (3\pi-4)^2$

How do you arrive at the result $$I=\displaystyle\int_{\mathbb{R^{+}}} \dfrac{\sin^3 {(\pi x^2)} \cos {(4x^2)}}{x^5} dx=\dfrac{\pi}{32} (3\pi-4)^2\ ?$$ Wolfram Alpha agrees numerically. I tried ...
2
votes
2answers
99 views

How to do integral $\int_0^{\infty} e^{-x^2-ax^4}\ dx , \ \text{ for $a>0$}$

I was told by this OP, $$\int_{0}^{\infty} e^{\large-x^n} \,dx =\Gamma \left(\frac{n+1}{n}\right), \qquad\text{ for $n>1$}.$$ This is via the variable change $t=x^n$: $$\int_{0}^{\infty} ...
0
votes
2answers
129 views

Improper integral of $\frac{\ln x}x$

Find $$\int_e^{\infty}\frac{\ln x}{x}\ dx$$ $A.\ \dfrac12$ $B.\ \dfrac{e^2}{2}$ $C.\ \dfrac{\ln(2e)}{2}$ $D.$ DNE (Does not exist) I tried doing this and this is where I've gone so far: $$\lim ...
1
vote
3answers
133 views

Evaluate $\int_0^\infty\frac{dl}{(r^2+l^2)^{\frac32}}$

How to evaluate the following integral $$\int_0^\infty\frac{dl}{(r^2+l^2)^{\large\frac32}}$$ The solution is supposed to look like this, unfortunately I can't derive it. $$ ...
1
vote
3answers
138 views

Value of the integral $\int_{\mathbb{R}} \frac{x\sin {(\pi x)}}{(1+x^2)^2}$

How do we evaluate the integral $$I=\displaystyle\int_{\mathbb{R}} \dfrac{x\sin {(\pi x)}}{(1+x^2)^2}$$ I have wasted so much time on this integral, tried many substitutions $(x^2=t, \ \pi x^2=t)$. ...
9
votes
2answers
185 views

An exercise from my brother: $\int_{-1}^1\frac{\ln (2x-1)}{\sqrt[\large 6]{x(1-x)(1-2x)^4}}\,dx$

My brother asked me to calculate the following integral before we had dinner and I have been working to calculate it since then ($\pm\, 4$ hours). He said, it has a beautiful closed form but I doubt ...
4
votes
3answers
160 views

Evaluate $\int_0^1\frac{x^a-x^{-a}}{x-1}dx$

I have heard that: $$\int_0^1\frac{x^a-x^{-a}}{x-1}dx=\frac1 a-\pi\cot(\pi a)$$ when $-1<a<1$. How would I prove this? That doesn't have an elementary indefinite integral, but the definite ...
4
votes
2answers
91 views

Evaluate $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$

I need to evaluate the following integral: $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$$ I thought of evaluating the iterated integral ...
21
votes
1answer
359 views

Prove ${\large\int}_0^\infty\frac{\ln x}{\sqrt{x}\ \sqrt{x+1}\ \sqrt{2x+1}}dx\stackrel?=\frac{\pi^{3/2}\,\ln2}{2^{3/2}\Gamma^2\left(\tfrac34\right)}$

I discovered the following conjecture by evaluating the integral numerically and then using some inverse symbolic calculation methods to find a possible closed form: $$\int_0^\infty\frac{\ln ...