Calculus of a Single Variable, Chapter 8, 8.1, Section 8.1, Problem 54

(dr)/dt=(1+e^t)^2/e^(3t)
r=int(1+e^t)^2/e^(3t)dt
r=int(1+2(1)e^t+(e^t)^2)/e^(3t)dt
r=int(1+2e^t+e^(2t))/e^(3t)dt
r=int(1/e^(3t)+(2e^t)/e^(3t)+e^(2t)/e^(3t))dt
r=int(e^(-3t)+2e^(t-3t)+e^(2t-3t))dt
r=int(e^(-3t)+2e^(-2t)+e^(-t))dt
Apply the sum rule and take the constants out,
r=inte^(-3t)dt+2inte^(-2t)dt+inte^(-t)dt
Now use the common integral: inte^x=e^x
r=e^(-3t)/(-3)+2e^(-2t)/(-2)+e^(-t)/(-1)
simplify and add a constant C to the solution,
r=-1/3e^(-3t)-e^(-2t)-e^(-t)+C
r=-(1/3e^(-3t)+e^(-2t)+e^(-t))+C