Question

Evaluate the integral
${\int }_1^{e^3}\frac{dx}{x\sqrt{1+\ln x}}$

• A. $2$
• B. $2\sqrt{2}$
• C. $\sqrt{2}$
• D. $-2$

Answer 1

The correct option is A $2$

${\int }_1^{e^3}\frac{dx}{x\sqrt{1+lnx}}={\int }_1^{e^3}\frac{dlnx}{\sqrt{1+lnx}}$

${\int }_1^{e^3}\frac{dlnx}{\sqrt{1+lnx}}=2\sqrt{1+lnx}{|}_1^{e^3}$

$=2\sqrt{1+3}-2\sqrt{1}$

$=2\sqrt{4}-2=2\left(1\right)$

$=2$

So, ${\int }_1^{e^3}\frac{dx}{x\sqrt{1+lnx}}=2$