Linked Questions

0
votes
1answer
121 views

Evaluation of $\int^{1}_{0}\bigg(\frac{1}{1-x}+\frac{1}{\ln x}\bigg)dx$ [duplicate]

Evaluation of $$\int^{1}_{0}\bigg(\frac{1}{1-x}+\frac{1}{\ln x}\bigg)dx$$ Let $$I = \int^{1}_{0}\bigg(\frac{1}{1-x}+\frac{1}{\ln x}\bigg)dx = \int^{1}_{0}\frac{(1-x)+\ln x}{(1-x)\ln x}dx$$ Now How ...
40
votes
6answers
1k views

Computing $ \int_0^\infty \frac{\log x}{\exp x} \ dx $ [duplicate]

I know that $$ \int_0^\infty \frac{\log x}{\exp x} = -\gamma $$ where $ \gamma $ is the Euler-Mascheroni constant, but I have no idea how to prove this. The series definition of $ \gamma $ leads me ...
17
votes
4answers
3k views

Integral representation of Euler's constant

Prove that : $$ \gamma=-\int_0^{1}\ln \ln \left ( \frac{1}{x} \right) \ \mathrm{d}x.$$ where $\gamma$ is Euler's constant ($\gamma \approx 0.57721$). This integral was mentioned in Wikipedia as in ...
4
votes
5answers
311 views

How do we prove that $\int_{0}^{1}x^n\left({1\over \ln{x}}+{1\over 1-x}\right)dx={\gamma+\ln(1+n)-H_n}$

How do we prove that n $\in$ $\Re$ $$\int_{0}^{1}x^n\left({1\over \ln{x}}+{1\over 1-x}\right)dx=\color{red}{\gamma+\ln(1+n)-H_n}.\tag1$$ $$J=\int_{0}^{1}{x^n\over 1-x}dx=\sum_{k=0}^{\infty}(-1)^k\...
5
votes
3answers
342 views

How do we prove that $\int_0^1 \ln x\left({1\over \ln{x}}+{1\over 1-x}\right)^2\,dx=\gamma-1?$

How do we prove that: $$\int_{0}^{1}\ln{x}\left({1\over \ln{x}}+{1\over 1-x}\right)^2\, dx =\color{blue}{\gamma-1}?\tag1$$ The only idea came to mind was this series $$\sum_{n=1}^{\infty}{1\over ...
10
votes
2answers
326 views

Show a detail prove of : $\int_{0}^{1}\int_{0}^{1}\left({x\over 1-xy}\cdot{\ln{x}-\ln{y}\over \ln{x}+\ln{y}}\right)\mathrm dx\mathrm dy=1-2\gamma$

Variation of my recent post. Strangely it leads to the result in term of Euler's constant;$\gamma$ Prove that $$\int_{0}^{1}\int_{0}^{1}\left({x\over 1-xy}\cdot{\ln{x}-\ln{y}\over \ln{x}+\ln{y}}\...
3
votes
4answers
209 views

How can one show that $\int_{0}^{\infty}\left({1\over 1+nx^n}-e^{-nx^n}\right)\cdot{\mathrm dx\over x^{1+n}}=1-\gamma?$

Consider $$\int_{0}^{\infty}\left({1\over 1+nx^n}-e^{-nx^n}\right)\cdot{\mathrm dx\over x^{1+n}}=1-\gamma\tag1$$ $n\ge1$;(integers) n seem to be not involved in the closed form(why?) How does ...
1
vote
4answers
121 views

Evaluate $\int_{0}^{1}\left [ \frac{1+\sqrt{1-x}}{x} +\frac{2}{\ln\left ( 1-x \right )}\right ]\, \mathrm{d}x$

Evaluate $$\int_{0}^{1}\left [ \frac{1+\sqrt{1-x}}{x} +\frac{2}{\ln\left ( 1-x \right )}\right ]\, \mathrm{d}x$$ I tried to let $1-x\rightarrow x$ ,and got $$\int_{0}^{1}\left [ \frac{1+\sqrt{1-x}}...
3
votes
1answer
101 views

Evaluating $\int_0^\infty (\operatorname{E}_n(x)e^x-\frac1{1+x})dx$

I want to evaluate $$I_n=\int_0^\infty \left(\operatorname{E}_n(x)e^x-\frac1{1+x}\right)dx=-\psi(n)$$where $\operatorname{E}_n$ denotes exponential integral and $\psi$ denotes polygamma function. ...
1
vote
1answer
164 views

Integral representation of Euler-Mascheroni constant

I'm solving an exercise in complex analysis. I proved $\displaystyle \gamma=\int_0^1 -\log(\log(1/s)) \, ds$. I want to prove that $\displaystyle \gamma=\int_0^1 \frac{1-e^{-t}-e^{-1/t}} t \,dt$. ...
1
vote
1answer
164 views

How to solve the integral $I=\int_0^\infty (e^{-u}\log u) du$ using standard methods [duplicate]

The following question from $\textit{Statistical Inference }$by Casella & Berger was given to me on an assignment last semester. On the surface, it appeared to be a straight-forward question about ...
1
vote
0answers
49 views

A collection of definite integrals with their derivations

Is there any book which includes proofs of the following integrals (and perhaps many other important definite integrals)?