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We have an obvious first step in simplifying this integral for solution: we factor the denominator of the integrand and obtain

(Eq'n 1)

If we had an integrand with only one of the factors in the denominator, we would have a trivial integration to solve. In this integration the special trick consists of noticing that we can contrive just that situation. If we multiply and divide the integral by 2a and undo the implicit sum, we get

(Eq'n 2)

But lnA - lnB = ln(A/B), so we have at last

(Eq'n 3)

Example Application:

Relativistic Astrogation

    Beginning at rest in the inertial frame of reference occupied and marked by the sun (approximately), a starship accelerates at a uniform rate A (as measured by the crew) for a time T measured on the ship's clocks. We want to know its velocity relative to the sun when the time T has elapsed. Using the relativistic velocity addition formula, the ship's astrogator calculates the change in the sun's velocity relative to the ship as

(Eq'n 4)

(The denominator drops out in the last step because vAdT/c2<<1). We then have the integration

(Eq'n 5)

We end up with

(Eq'n 6)

We can multiply that equation by two, take the antilogarithm, and then solve for v/c to get

(Eq'n 7)


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