Question

y=x^2e^x find the indefinite integral funtion

Answer 1

Let's integrate $\int x^2e^xdx$ by parts. Let's use the rule:

$$\int fg'=fg-\int f'g,$$

here, $f=x^2, g=e^x$.

Then, we get:

$$\int x^2e^xdx=x^2e^x-2\int xe^xdx.$$

Let's integrate $\int xe^xdx$ by parts and use the same rule. This time $f=x, g=e^x$. Finally, we get:

$$\int x^2e^xdx=x^2e^x-2(xe^x-\int e^xdx),$$ $$\int x^2e^xdx=x^2e^x-2xe^x+2e^x=(x^2-2x+2)e^x+C.$$