<s>[INST] Find the exact value of integration $ \int_0^1 \frac{1}{\sqrt{1-x}+\sqrt{x+1}+2} dx$ Can you help me find the exact value for integration with the given steps?
$$ \int_0^1 \frac{1}{\sqrt{1-x}+\sqrt{x+1}+2} dx$$
Some of my attempts as indefinite Integral
$$
\int \frac{1}{2+\sqrt{1-x}+\sqrt{1+x}} \, dx\approx \left(\sqrt{x+1}+\left(-\frac{1}{\sqrt{x+1}+1}-1\right) \sqrt{1-x}+\frac{1}{\sqrt{x+1}+1}-\frac{2 \left(0.707107 \sqrt{x+1}\right)}{\sin }\right)+C
$$ 
Is it considered Improper Integral?
 [/INST] Substitute $x = \sin 2t $ to have
$$\sqrt{1-x} = \cos t - \sin t, \>\>\>\>\>\sqrt{1+x} = \cos t + \sin t$$
and,
$$\begin{align}
& \int_0^1 \frac{dx}{\sqrt{1-x}+\sqrt{x+1}+2} \\
& = \int_0^{\pi/4}\frac{\cos2t}{1+\cos t}dt \\
&= \int_0^{\pi/4}\frac{2(1+\cos t)^2 -4 (1+\cos t) +1}{1+\cos t}dt \\
& =\int_0^{\pi/4} \left(-2 + 2\cos t + \frac12\sec^2 \frac t2\right)dt \\
&= \left(-2t + 2\sin t + \tan \frac t2\right)\bigg|_0^{\pi/4}= 2\sqrt2-1-\frac\pi2
\end{align}$$
</s>