We have #intt^2e^(4t)dt#.
According to Integration by Parts, #intf(t)g(t)dt=f(t)intg(t)dt-intf'(x)(intg(t)dt)dt#
Here, #f(t)=t^2# and #g(t)=e^(4t)#. So we input:
#t^2inte^(4t)dt-int(t^2)'(inte^(4t)dt)dt#
A logical route to take is to find #inte^(4t)dt#
According to Integration by Substitution, #intf(g(t))g'(t)dt=intf(u)du#, where #u=g(t)#. We can write the above as:
#1/4inte^(4t)4dt#
#1/4inte^(u)du#
#1/4e^u#
#e^(4t)/4#. So we input:
#(t^2e^(4t))/4-1/2intte^(4t)dt#
Apply integration by parts for the integral:
#tinte^(4t)dt-intt'(inte^(4t)dt)dt#
Since we know that #inte^(4t)dt# is:
#(te^(4t))/4-inte^(4t)/4dt#
#(te^(4t))/4-1/4*1/4e^(4t)#
#(te^(4t))/4-e^(4t)/16#
We can input this into our eariler calculations:
#(t^2e^(4t))/4-1/2((te^(4t))/4-e^(4t)/16)#
Add the constant of integration:
#(t^2e^(4t))/4-1/2((te^(4t))/4-e^(4t)/16)+C#
