The "integration by parts" method can be seen as a form of inverse product rule for differentiation:
int fg' dx = fg - int f'g dx, where f,g functions of x.
Here, we see that (e^x)' = e^x, so this matches directly with
int fg' dx where f(x) = x^2 and g(x) = e^x.
Therefore,
int x^2e^xdx = x^2e^x - int 2xe^xdx
Now, since e^x can be differentiated indefinitely without a change, we can apply this method again, differentiating repeatedly until the integrals are gone:
int 2xe^xdx = 2xe^x - int 2e^xdx
=2xe^x -2inte^xdx
=2xe^x - 2e^x + c
Now, we have to subtract this from the original x^2e^x (the fg part in our example) to get the final answer:
intx^2e^xdx = x^2e^x -2xe^x + 2e^x +c = e^x(x^2 - 2x + 2) + c
I did realize I said subtract, yet still put +c instead of -c at the end. That is because it's an arbitrary constant, and can be any real number, and in the end, most people use +c in their notation.
