Question

${\int }_1^2\frac{dx}{\sqrt{1+x^2}}=$

• A. ${\log }_e\left(\frac{2+\sqrt{5}}{\sqrt{2}+1}\right)$
• B. ${\log }_e\left(\frac{\sqrt{2}+1}{2+\sqrt{5}}\right)$
• C. ${\log }_e\left(\frac{2-\sqrt{5}}{\sqrt{2}-1}\right)$
• D. 0

Answer 1

Answer: (A)

${\log }_e\left(\frac{2+\sqrt{5}}{\sqrt{2}+1}\right)$

${\int }_1^2\frac{dx}{\sqrt{1+x^2}}=\log |x+\sqrt{x^2+a}|+c$
${\int }_1^2\frac{dx}{\sqrt{1+x^2}}=\log |x+\sqrt{x^2+1}|+c$
$=\log |2+\sqrt{5}|-\log |1+\sqrt{2}|$
$=\log \left|\frac{2+\sqrt{5}}{1+\sqrt{2}}\right|$
${\int }_1^2\frac{dx}{\sqrt{1+x^2}}=\log \left|\frac{2+\sqrt{5}}{1+\sqrt{2}}\right|$