Question

How to integrate this:
Integrate(x^2*e^x) with dx , starting at 0 to 1?

Answer 1

Integrate by parts twice.

Answer 2

put
\[
u =x^2; \quad dv=e^x dx\\
du= 2 x; v= e^x\\
\int x^2 e^x dx = x^2 e^x - \int 2 x e^x dx
\]

Answer 3

For the last integral
put
\[
u=2x ;\quad dv=e^x\\
du= 2 dx ; \quad v=e^x

\int 2 x e^x dx = 2 xe^x - \int 2 e^x dx=\\
2 xe^x - 2 e^x
\]

Answer 4

Combining and simplifying, you get
\[
\int x^2 e^x \, dx=e^x \left(x^2-2
x+2\right)
\]

Answer 5

\[
\int_0^1 x^2 e^x \, dx=\left[e^x \left(x^2-2
x+2\right)\right]_0^1=-2 +e
\]

Answer 6

Thanks!