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For x>0x > 0x>0, define f(x)f(x)f(x) as follows: f(x)=∫0x(t2+1−t) dt.f(x) = \int_{0}^{x} \left( \sqrt{t^2+1} - t \right) \, dt.f(x)=∫0x(t2+1−t)dt. Let L=limx→∞(f(x)−lnx2).L = \displaystyle\lim_{x\to\infty} \left(f(x) - \dfrac{\ln x}{2}\right).L=x→∞lim(f(x)−2lnx).
If L=a+lnbcL = \dfrac{a + \ln{b}}{c}L=ca+lnb for positive integers a,b, a, b,a,b, and ccc with gcd(a,c)=1 \gcd(a,c) = 1 gcd(a,c)=1, then find the value of a+b+c a + b + c a+b+c.
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