# This integral does not converge as $$x \to \infty$$

Calculus Level 5

For $$x > 0$$, define $$f(x)$$ as follows: $f(x) = \int_{0}^{x} \left( \sqrt{t^2+1} - t \right) \, dt.$ Let $$L = \displaystyle\lim_{x\to\infty} \left(f(x) - \dfrac{\ln x}{2}\right).$$

If $$L = \dfrac{a + \ln{b}}{c}$$ for positive integers $$a, b,$$ and $$c$$ with $$\gcd(a,c) = 1$$, then find the value of $$a + b + c$$.

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