First, Integrate.
So, #int (2x - xe^x)(-xe^x + secx)dx#
#= int (-2x^2e^x -x^2e^(2x) + 2xsec x - xe^xsecx)dx#
#= -2int x^2e^xdx -int x^2e^(2x)dx +2 int xsecx dx -int xe^xsecxdx#
Now We have to Integrate By Parts.
Assume #I_1 = -2 int x^2e^xdx#,
#I_2 = -int x^2e^(2x)dx#
#I_3 = 2int x sec xdx#
#I_4 = -intxe^xsecxdx#
Now The Problem is,
#I_1# and #I_2# can be found, but #I_3# and #I_4# can't be expressed by elementary functions. These integrals are undeterminable.
So, Get the #I_1# first.
#I_1 = -2[x^2inte^xdx - int {d/dx(x^2)inte^x dx}dx]#
#= -2 [x^2e^x - 2 intxe^xdx]#
#= -2[x^2e^x - 2{x inte^xdx - int (d/dx(x) int e^xdx)dx}]#
#= -2[x^2e^x - 2{xe^x - inte^xdx}]#
#= -2[x^2e^x - 2{xe^x -e^x}]#
#= -2[x^2e^x - 2xe^x +2e^x]#
#= -2e^x[x^2 - 2x + 2]#
Now, #I_2#.
#I_2 = -2[x^2inte^(2x)dx - int {d/dx(x^2)inte^(2x) dx}dx]#
#= -2 [1/2x^2e^(2x) - 2 intxe^(2x)dx]#
#= -2[1/2x^2e^(2x) - 2{x inte^(2x)dx - int (d/dx(x) int e^(2x)dx)dx}]#
#= -2[1/2x^2e^(2x) - 2{1/2xe^(2x) - inte^(2x)dx}]#
#= -2[1/2x^2e^(2x) - 2{1/2xe^(2x) -1/2e^(2x)}]#
#= -2[1/2x^2e^(2x) - xe^(2x) +e^(2x)]#
#= -2e^(2x)[1/2x^2 - x + 1]#
So, The Entire Integral is :
#int(2x - xe^x)(-xe^x + sec x) dx#
#= -2e^x[x^2 - 2x + 2] -2e^(2x)[1/2x^2 - x + 1] + 2 intxsecxdx - intxe^xsecxdx#
Hope this helps, but It won't, most probably.
