Question 564912
*[tex \LARGE \int 3x^2e^{-x} dx]


Begin by letting u = 3x^2 and dv = e^(-x) dx. Then du = 6xdx and v = -e^(-x). Applying integration by parts,


*[tex \LARGE \int 3x^2e^{-x} dx = -3x^2e^{-x} - \int -6xe^{-x}dx = -3x^2e^{-x} + \int 6xe^{-x}dx]


Again, we cannot really integrate 6xe^(-x) dx without using parts, so we let y = 6x, dz = e^(-x)dx --> dy = 6dx and z = -e^(-x). Hence,


*[tex \LARGE -3x^2e^{-x} + \int 6xe^{-x}dx = -3x^2e^{-x} + (-6xe^{-x} - \int -6e^{-x} dx)]


And just carry it out from there. Remember to add a constant when you are done, since antiderivatives are not uniquely determined and can differ by a constant.