To determine the number of textbooks sold in the first three years, we need to integrate the given function from 0 to 3 years.
The formula for integration by parts is:
∫u dv = uv - ∫v du
Let's denote:
u = t
dv = 30,000e^(-1.5t) dt
To find du and v, we take the derivatives and antiderivatives, respectively:
du = dt
v = ∫30,000e^(-1.5t) dt
To find v, we can use the substitution method:
Let u = -1.5t, then dv = 30,000e^u du
∫30,000e^(-1.5t) dt = v = ∫e^u du
Integrating e^u, we have:
v = -20,000e^(-1.5t)
Substituting the values back into the integration by parts formula, we have:
∫(t)(30,000e^(-1.5t)) dt = (t)(-20,000e^(-1.5t)) - ∫-20,000e^(-1.5t) dt
Now, we can integrate the second term on the right-hand side:
∫-20,000e^(-1.5t) dt = -20,000 ∫e^(-1.5t) dt
To find this integral, we can once again use substitution:
Let u = -1.5t, then du = -1.5 dt
So, the integral becomes:
-20,000 ∫e^u(-1.5) du = 30,000e^(-1.5t)
Now, we substitute the values back into the integration by parts formula:
(t)(-20,000e^(-1.5t)) - ∫-20,000e^(-1.5t) dt
= -20,000te^(-1.5t) + 30,000e^(-1.5t)
Evaluating this expression from t = 0 to t = 3 gives us the total number of textbooks sold in the first three years:
(-20,000(3)e^(-1.5(3)) + 30,000e^(-1.5(3))) - (-20,000(0)e^(-1.5(0)) + 30,000e^(-1.5(0)))
Simplifying this expression gives us the final answer.