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I am trying to understand this equality:

\ln{\left|\frac{x}{2}+\sqrt{\frac{x^2}{4}+1}\right|} + C= \ln{|x+\sqrt{x^2+4}|} + C'

My teacher didn't really explain it, she just noted that "the difference between the two statements is a constant (This equality is an answer for an integral so she just changed $C$ to $C'$).

Can anyone please explain it? Thanks!

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    Hint: $\ln(x)=\ln(2*x)-\ln(2)$2012-01-15

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$\ln{\left|\frac{x}{2}+\sqrt{\frac{x^2}{4}+1}\right|} + C=\ln{\left|\frac{x}{2}+\sqrt{\frac{x^2+4}{4}}\right|}+C$ $=\ln{\left|\frac{x+\sqrt{x^2+4}}{2}\right|}+C$ $= \ln{|x+\sqrt{x^2+4}|} -\ln2+ C$ = \ln{|x+\sqrt{x^2+4}|}+ C'.