Question

The value of the definite integral ${\int }_1^e((x+1)e^x.\ln x)dx$ is

• A. $e$
• B. $e^{e+1}$
• C. $e^e(e-1)$
• D. $e^e(e-1)+e$

Answer 1

Answer: (D)

$e^e(e-1)+e$

$I={\int }_1^e(x+1)e^x\iota nxdx={\int }_1^e[x\iota nx+\iota nx+1-1]e^{x}dx={\int }_1^e[x\iota nx+\iota nx+1]e^{x}dx-{\int }_1^e{e}^{x}dx={\left[x\iota ne{x}^x\right]}_e^1-{\left[e^x\right]}_e^1=e{e}^e-e^e+e$