[tex] \int\limits^0_1 {x^2e^{2x}} \, dx = \\ x^2 \frac{e^{2x}}{2}|_1^0 - \int\limits^0_1 {2x \frac{e^{2x}}{2} } \, dx= \\ x^2 \frac{e^{2x}}{2}|_1^0 - \int\limits^0_1 {xe^{2x}} \, dx= \\ x^2 \frac{e^{2x}}{2}|_1^0 -x \frac{e^{2x}}{2}|_1^0+ \int\limits^0_1 { \frac{e^{2x}}{2} } \, dx= \\ x^2 \frac{e^{2x}}{2}|_1^0 -x \frac{e^{2x}}{2}|_1^0+ \frac{e^{2x}}{4}|_1^0 = \\ \frac{e^{2x}}{4}(2x^2-2x+1)|_1^0= \frac{3}{4}- \frac{e^2}{4}= \frac{3-e^2}{4} [/tex]
