Question

Evaluate the integral
${\int }_0^1{x}^2{e}^{x}dx$

• A. $e-2$
• B. $e+2$
• C. $e$
• D. $e+3$

Answer 1

Answer: (A)

$e-2$

${\int }_0^1{x}^2{e}^{x}dx$

$\int x^2{e}^{x}dx=x^2{e}^x-2\int e^{x}dx+c$

$\int x{e}^{x}dx=x{e}^x-e^x+c$

So, $\int x^2{e}^{x}dx=x^2{e}^x-2[x{e}^x-e^x]+c$

$=e^x[x^2-2x+2]+c$

${\int }_0^1{x}^2{e}^{x}dx={\left[e^x\right[x^2-2x+2]+c]}_0^1$

$=\left(e^1\right[1-2+2]+c)-\left(1\right(0+2)+c)$

$=e\left(1\right)-2$

${\int }_0^1{x}^2{e}^{x}dx=e-2$