Question

The value of definite integral ${\int }_1^e\sqrt{x}ln\left(x\right)dx$ is

• A. $\frac{4}{9}\sqrt{e^3}+\frac{2}{9}$
• B. $\frac{2}{9}\sqrt{e^3}-\frac{4}{9}$
• C. $\frac{2}{9}\sqrt{e^3}+\frac{4}{9}$
• D. $\frac{4}{9}\sqrt{e^3}-\frac{2}{9}$

Answer 1

The correct option is C $\frac{2}{9}\sqrt{e^3}+\frac{4}{9}$

By using Integration by parts,
${\int }_1^0\sqrt{x}ln\left(x\right)dx={\left(\frac{2}{3}lnx{x}^{3/2}\right)}_1^e-{\int }_1^0\frac{1}{x}\left(\frac{2}{3}{x}^{3/2}\right)dx$

$=\frac{2}{3}{e}^{3/2}-\frac{4}{9}{\left(x^{3/2}\right)}_1^e$
$=\frac{2}{3}{e}^{3/2}-\frac{4}{9}{e}^{3/2}+\frac{4}{9}$
$=\frac{2}{9}\sqrt{e^3}+\frac{4}{9}$