'Almost rational' integrals with no known closed form? I recently stumbled upon an 'almost rational' integral, namely:
$$\int_0^{\pi/2} x \frac{\sqrt{\sin x}-\sqrt{\cos x}}{\sqrt{\cos x}+\sqrt{\sin x}} dx=0.231231222\dots \approx 0.231231231\dots= \frac{77}{333}$$
The error here is only:
$$\frac{77}{333}-I \approx 9 \cdot 10^{-9}$$
I don't think this integral has any known closed form (although I could be mistaken).

Do you know any integrals without a known closed form which are 'almost rational', meaning their value can be approximated by a rational number with the conditions:

$$|I-\frac{p}{q}|<10^{-7}$$
$$|p|,|q|<500$$
I hope there are some non-trivial integrals like the example I provided.
I tried to be as clear as possible, because the MathWorld page about almost integers is filled with 'questionable' entries.
 A: One of the most remarkable integrals for me is the Borwein sequence. They do have closed form, but somehow I feel they should be mentioned here.
More interesting integrals.
A: At least your integral turns out to have a closed form:
$$ \int_{0}^{\pi/2} x \, \frac{\sqrt{\sin x} - \sqrt{\cos x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, \mathrm{d}x = G + \pi \left( \frac{1+2\sqrt{2}}{4} \log 2 - \log (1+\sqrt{2}) \right), $$
where $G$ is the Catalan's constant. So it seems to me that the 'almost rationality' of this integral is just a coincidence.
A: Integral of $e^{-2x^2}$ from $-2$ to $62$ is about $1/8$ which has a  $.003$ error. Also maybe cheating but its possible to get this integral arbitrarily small giving really small errors to something without closed form. This has implication that you could integrate any probability distributions most of which have no close form to 1 plus an arbitrary small error. 
Integral of $e^{-2x^2}$  from $2(e^{\pi}-\pi)/100$ to $2(e^{\pi}-\pi)$ = .5 with .006 error
A: Consider any function $f$ such that $\int_{a}^{b} f(x)dx$ does not have a closed form, now choose $b-a$ small enough that the integral is very close to zero.
Eg: $\int_{0}^{1} e^{-x^{100}} dx$ perhaps does not have closed form and is surely very close to zero.
Any such function can be modified to make the integral very close to any given rational integer.
