Question

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

Answer 1

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

$=x^2{e}^2-|2x{e}^2-\frac{e^{2x}}{2}$

$=x^2{e}^2-2\left[x{e}^2-|e^x\right]-\frac{1}{2}{e}^{2x}$

$=x^2{e}^2-2\left(x{e}^x-e^x\right)-\frac{1}{2}{e}^{2x}{{|}^1}_{x=0}$

$=\left(e-2\left(e-e\right)-\frac{1}{2}{e}^2\right)-\left(0+2-\frac{1}{2}\right)$

$=e-\frac{1}{2}{e}^2-\frac{3}{2}$