The following formula (from my integration sheets provided from college): $$\int\frac{dx}{\sqrt{a^2+x^2}}=\ln ({x+\sqrt{a^2+x^2}})+C$$ appears that it doesn't hold for this classic physics problem: A particle is moving along a trajectory with acceleration $\frac{1}{\sqrt{4+t^2}}$ $\frac{m}{s^2}$. Find the velocity of the particle at $t=1$ second, if its initial velocity (at $t=0$) is $v=5\:\frac{m}{s}$. According to the formula the result should simply be $$\ln\:({1+\sqrt{4+1}})+5\approx6.17\:\frac{m}{s}$$ The result we obtained should be incorrect. If we dig deeper into the actual integral and solve it using the hyperbolic substitution $(x=u\sinh t)$, and identity $\cosh^2 t-\sinh^2 t=1$ we obtain the following $$\int\frac{dx}{\sqrt{a^2+x^2}}=\ln \left( \frac{x}{a}+\sqrt{1+\frac{x^2}{a^2}} \right)+C$$ And if we now calculate the solution using this formula, the solution is correct: $$v\approx5.48\:\frac{m}{s}.$$ I know that we can factor out $\frac{1}{a}$ and get $$\int\frac{dx}{\sqrt{a^2+x^2}}=\ln \left(\frac{1}{a} \left( x+\sqrt{a^2+x^2} \right)\right)+C.$$ and then rewrite this as a sum of two natural logarithms from which one is a constant and get the exact equation that we started with in the first place, but exactly this step produces the error into this physics problem.
*Now, the question is, since the formula must hold for every integration (and thus this physics problem) isn't the provided formula literally wrong? Since it must hold in every situation, factoring out $\frac{1}{a}$ and ultimately rewriting it as the sum of two logarithms and joining it with a constant would produce an error in some tasks so, logically, this step, although mathematically correct, for a formula can't be done?
Apply the integral result to the velocity
$$v(t) = \int\frac{dt}{\sqrt{4+t^2}}=\ln ({t+\sqrt{4+t^2}})+C$$
Substitute the initial condition $v(0) = 5$ to obtain the constant $C$
$$5 =\ln (2)+C\implies C = 5-\ln(2)$$
Therefore,
$$v(t)=\ln ({t+\sqrt{4+t^2}})-\ln(2)+5$$
Then, evaluate $v(t)$ for $t=1$,
$$v(1)=\ln ({1+\sqrt{4+1}})-\ln(2)+5=5.48 \text{ m}/\text{s}$$
which gives the correct result.