How do you use integration by parts to find #intxe^x dx#?

1 Answer
Oct 7, 2014

To integrate by parts, we have to pick a #u# and #dv# such that

#int u dv= uv-int v du#

We can pick what #u# and #dv# are, though we should typically try to pick #u# such that #du# is "simpler". (By the way, #du# just means the derivative of #u#, and #dv# just means derivative of #v#)

So, from #int u dv#, let's pick #u=x# and #dv=e^x#. It is helpful to fill in all of the parts you will use throughout the problem before you start integrating:

#u=x#
#(du)/dx=1# so #du=1dx=dx#

#dv=e^x#
#v=int dv= int e^x dx =e^x#

Now let's plug everything back into the original formula.

#int u dv= uv-int v du#
#int xe^x dx= xe^x-int e^xdx#
You can integrate the last part quite simply now, to get:

#int xe^x dx= xe^x-int e^xdx= xe^x-e^x +c#