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
