This integral does not converge as xx \to \infty

Calculus Level 5

For x>0x > 0, define f(x)f(x) as follows: f(x)=0x(t2+1t)dt.f(x) = \int_{0}^{x} \left( \sqrt{t^2+1} - t \right) \, dt. Let L=limx(f(x)lnx2).L = \displaystyle\lim_{x\to\infty} \left(f(x) - \dfrac{\ln x}{2}\right).

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

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