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    We have three ways in which we can solve this integral. I want to present all three of them for the insight that they may offer into the integration of exponential functions.

    1. The simplest of the three is an integration by parts. We have the standard  ∫udv=uv-∫vdu, in which u=x and dv=exdx. Thus we have

(Eq'n 1)

    2. In the other integration by parts we have u=ex and dv=xdx, so we get

(Eq'n 2)

We know that

(Eq'n 3)

so if we add/subtract the term ex(1-x) to/from Equation 2, we get

(Eq'n 4)

If we absorb the one into the implicit constant of integration, we get the same result that we got in Equation 1.

    3. Finally we have the term-by-term integration of the infinite series representation of the integrand, in which we exploit the fact that

(Eq'n 5)

We have, then, for our integral

(Eq'n 6)

We then repair the coefficient,

(Eq'n 7)

Using that, we transform Equation 6 into

(Eq'n 8)

And again we absorb the one into the implicit constant of integration and recover Equation 1.

Example Application:

Resource Depletion

    For at least several centuries the human population of Earth has been growing exponentially. If we pick a certain date t0 at which the population was P0, then at a later time t1 the population is

(Eq'n 9)

in which alpha represents the growth rate of the population and t=t1-t0. In the same period the average per person use of a certain resource (e.g. iron, copper, petroleum, rubber, etc.) has increased more or less linearly; that is, the per capita amount of that resource used per time interval conforms to

(Eq'n 10)


in which A0 represents the rate of resource use at time t0 and beta represents the rate at which that use increases. The amount of the resource used up in that time period thus equals

(Eq'n 11)

Of course, that formula is too simplistic to provide a proper model of Reality. But it gives us a good starting point for developing a proper theory describing how population growth and social development affect Humanity's use of resources.


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