1
$\begingroup$

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!

  • 3
    Hint: $\ln(x)=\ln(2*x)-\ln(2)$2012-01-15

1 Answers 1

3

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