$$ \pi = \int_{-\sqrt{2}}^{\sqrt{2}} \sqrt{2-x^2} dx $$
So, thinking I was going to discover something amazing I reasoned that this is equal to pi, and that all I had to do to get the exact value of pi was to find the indefinite integral.
$$ \int \sqrt{2-x^2} dx = \space ? $$
Of course, I couldn't, not even with partial integrals.
So now I'm wondering whether this is possible, but I don't think so.
:)
On
As an alternative to trigonometric substitution, you can also argue that this integral is equal to $\pi$ geometrically. As you probably are aware, the definite integral of a non-negative function $f$ on $[a,b]$ is the area of the region bounded by the curves $y = f(x)$, $y = 0$, $x = a$, and $x = b$. For $f(x) = \sqrt{2 - x^2}$, the curve $y = f(x)$ with $-\sqrt 2 \leq x \leq \sqrt 2$ is the semicircle $\{(x,y) \mid x^2 + y^2 = 2 \,\&\, y \geq 0 \}$. So your integral equals half the area within the whole circle $x^2 + y^2 = 2$, which is simply $\frac{1}{2}(2\pi) = \pi$.
Use trigonometric substitution, with $x = \sqrt 2 \sin\theta$.
Then $\,dx = \sqrt 2 \cos\theta\,d\theta$.