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This seems a fairly straightforward integration to carry out. We need only convert the integrand into the equivalent infinite series in powers of x and integrate the series term by term. We begin by factoring out 1/a and making the replacement bx/a = y to obtain

(Eq'n 1)

Multiplying and dividing that equation leaves it unchanged and gives it to us in the form

(Eq'n 2)

which mathematicians call the Mercator series with the proviso that |y|<1.

We can invert the infinite series

(Eq'n 3)

and obtain for its initial terms

(Eq'n 4)

By induction we infer that the remaining terms follow the same pattern that we discern in those, so that we have

(Eq'n 5)

Solving that equation for z gives us

(Eq'n 6)

We can absorb ln(a) into the tacit constant of integration, so we have at last

(Eq'n 7)

Example Application:

Entropy of a Heated Ideal Solid

Imagine that we have a solid body that has a heat capacity that we represent with C such that any increment of heat that we add to or take out of the body produces a change dT in the body's absolute temperature in accordance with

dQ=CdT.

(Eq'n 8)

We specify further that the value of C does not depend upon the body's temperature, but remains constant as the body heats up or cools down.

We want to calculate the entropy of the heat held by the body at a certain temperature. However, we can only measure temperature in degrees Celcius (Tc) while we must express the temperature in the laws of thermodynamics in degrees Kelvin (TK). The increment by which the entropy of the heat in the body changes when the body has temperature TK=273.15+Tc conforms to

(Eq'n 9)

To obtain a description of the entropy change in the body as its temperature changes from Tc1 to Tc2 we integrate that equation;

(Eq'n 10)

in accordance with Equation 7.

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