This integral has bothered me for the longest time: $$J=\int_{-1}^0 \sqrt[x]{2+\Gamma(x+1)}\space\text{dx}$$
This guy is extremely minuscule in relation to most other integrals but was amazingly difficult nonetheless. I ended up getting a reasonable solution, an ugly decimal form, which I checked with Wolfram Alpha to be true. That closed form is the following:
$$J=0.0496407...$$
But WA also has no closed form. As I am a student, I really don't have that much time to look at these things (I have been kind of teaching myself Calculus). I have made very many attempts, but the fact that I can't really break up that portion (because of the + sign), has thrown me for a loop. T I just really don't want to try too hard on this problem as I have homework and weight training later. Thanks all in advance.
There is no closed form, because For all $k\in\Bbb{R}$
$$\int_{-1}^0 (k+(\Gamma(x+1)))^{1/x}dx$$ has no closed form.