Integration by Parts
Derivation of Integration by Parts
Recall the product rule:
(uv)' = u' v + uv'
uv' = (uv)' - u' v
Integrating both sides, we have that
uv' dx = (uv)' dx - u' v dx
= uv - u' v dx.
We use integration by parts. Notice that we need to use substitution to find the integral of ex.
Hence we have
x ln x dx
Integration By Parts Twice
x2 ex dx
We use integration by parts
x2ex - 2xex dx = x2ex - 2xex dx
Have we gone nowhere? Now we now use integration by parts a second time to find this integral
x2ex - 2xex + 2ex dx
= x2ex - 2xex + 2ex + C
Other By Parts
Occasionally there is not an obvious pair of u and dv. This is where we get creative.
ln x dx
What should we let u and dv be? Try
x lnx - dx = x lnx - x + C
When to Use Integration By Parts
Evaluating a Definite Integral
We can use integration by parts to evaluate definite integrals. We just have to remember that all terms receive the limits.
Use integration by parts
Application: Present Value
Your patent brings you a annual income of 3,000 t dollars where t is the number of years since the the patent begins. The patent will expire in 20 years. A business has offered to purchase the patent from you. How much should you ask for it? Assume an inflation rate of 5%.
This question is a present value problem. Since there is inflation, your later earnings will be worth less than this year's earnings. The formula to determine this is given by
For our example, we have
c(t) = 2000 t r = 0.05 t1 = 20
Use integration by parts and note that with the substitution
u = -0.05t du = -0.05dt
-20du = dt
This gives us
We have already found the antiderivative for this last integral. We have
You should ask for $211,393.