int x^2 e^x dx
= x^2*e^x - int 2x*e^x
= x^2*e^x -( 2*x * e^x- int 2*e^x)
= x^2*e^x-(2*x * e^x - 2*e^x) +C
= e^x(x^2-2x+2)+C
Explanation:
Integrate by parts :
int x^2 e^x dx
Let u = x^2, v = e^x. You will get (d v) = e^x*dx, (du)=2x*dx, .
To reduce the power of x in the integral, you may rewrite it in this form int u * d v . This new expression is equivalent to the original one. (by replacing x^2 with u and e^x *dx with (d v).)
int u*d v = u*v - int du * v
int x^2 e^x dx = x^2*e^x - int 2x*e^x
Now, let's do the partial integration again with m = 2*x this time. In this case d m = 2 *dx
int m*d v = m*v - int dm * v
int 2x*e^x = 2*x * e^x- int 2*e^x = 2*x * e^x - 2*e^x +C
(include the arbritary constant C in the expression)
