Evaluate the following integral.
$$ \int x^2 e^x\ dx $$
What i have tried :
$ f^{'}(x) = e^x , f(x) = e^x , g(x) = x^2$
$$ \int x^2 e^x\ dx = e^x\ x^2 - \int e^x\ 2x\ dx $$
$ f^{'}(x) = e^x , f(x) = e^x , g(x) = 2x $
$$ \int x^2 e^x\ dx = e^x\ x^2 - e^x\ 2x - \int e^x\ x\ dx $$
$ f^{'}(x) = e^x , f(x) = e^x , g(x) = x $
$$ \int x^2 e^x\ dx = e^x\ x^2 - e^x\ 2x - x\ e^x - \int e^x \ dx $$
$$ \int x^2 e^x\ dx = e^x\ x^2 - e^x\ 2x - x\ e^x - e^x + C $$
The answer in the book is $ x^2e^x-2xe^x+2e^x+C $
What did i do wrong ?
In your third line,
$$ \int x^2 e^x\ dx = e^x\ x^2 - e^x\ 2x - \int e^x\ x\ dx $$
you should have:
$$\int x^2 e^x\ dx = e^x\ x^2 -\Big(e^x\ 2x - \int 2e^x\, dx\Big)$$
Then, integrating on the right gives and distributing the negative gives us $$x^2e^x - 2xe^x + 2e^x$$