Computing Indefinite Integrals — Question 1

Question 1

Compute the indefinite integral $$\int x^{2} e^{x}\,dx.$$

Question 1 - Solution

Use integration by parts with $$u=x^2, \qquad dv=e^{x}\,dx.$$ Then $$du=2x\,dx, \qquad v=e^{x}.$$ So $$\int x^2 e^{x}\,dx = x^2 e^{x} - \int 2x e^{x}\,dx.$$

Apply integration by parts again to $\int 2x e^{x}\,dx$: $$u=2x,\quad dv=e^{x}\,dx \Rightarrow du=2\,dx,\ v=e^{x}.$$ Thus, $$\int 2x e^{x}\,dx = 2x e^{x} - 2\int e^{x}\,dx = 2x e^{x} - 2e^{x}.$$

Hence, $$\int x^2 e^{x}\,dx = e^{x}(x^2 - 2x + 2) + C.$$

Final Answer: $$\boxed{ \int x^2 e^{x}\,dx = e^{x}(x^2 - 2x + 2) + C }$$