# Tagged Questions

175 views

### Can we express the following in a closed form? [duplicate]

I want to evaluate the integral: $$I=\int_{0}^{\pi/2}\ln \left ( \frac{(1+\sin x)^{1+\cos x}}{1+\cos x} \right )\,dx$$ Well, the sub $u=\pi/2-x$ does not give me any result. In fact it makes the ...
66 views

### Is there closed form for $(1-p)(1-p^2)(1-p^3)…$ or its Taylor expansion?

I was considering the following problem: Somebody uses a backup for something (e.g. backups a file) and the backup is equally reliable as original storage. The storage is not perfectly reliable and ...
267 views

### Formula for $\int_0^\infty \frac{\log(1+x^2)}{\sqrt{(a^2+x^2)(b^2+x^2)}}dx$

Is it possible to express the following integral in terms of known special functions? $$I(a,b)=\int_0^\infty \frac{\log(1+x^2)}{\sqrt{(a^2+x^2)(b^2+x^2)}}dx$$ I have managed to solve the special ...
176 views

### Closed Form for $\int_0^1 \frac{\log(x)}{\sqrt{1-x^2}\sqrt{x^2+2+2\sqrt{2}}}dx$

Is there a closed form for the following integral? $$\int_0^1 \frac{\log(x)}{\sqrt{1-x^2}\sqrt{x^2+2+2\sqrt{2}}}dx$$ It is approximately equal to $-0.48878092308456029189008$. Mathematica is ...
730 views

### A Challenging Integral $\int_0^{\frac{\pi}{2}}\log \left( x^2+\log^2(\cos x)\right)dx$

I encountered a strange integral with a strange result. $$\int_0^{\frac{\pi}{2}}\log \left( x^2+\log^2(\cos x)\right)dx = \pi \log \left(\log (2) \right)$$ Believe it or not, the result agrees ...