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We don't have an obvious antiderivative for this integrand, so we cannot integrate the form directly. We will have to develop the integrand and then integrate by parts. By develop the integrand I mean either multiply it by a formula that equals one (such as x/x), add to it a term that equals zero (such as x-x), or execute some combination of both. The presence of the variable of integration in the development gives the integrand a handle by which we can grasp the integration by parts.

If the numerator in the integrand were xdx instead of dx, then we could carry out the integration directly via an inversion of the chain rule. So we do the next best thing and multiply the integrand by x/x so we have

(Eq'n 1)

We integrate that improved integrand by parts by reversing the formula of compound differentiation, d(fg) = fdg+gdf, so we have

(Eq'n 2)

In the present case we have

(Eq'ns 3)

When we carry out the integration by parts we discover that we must repeat the process in an endless progression; the solution clearly comes to us as an infinite series. Using xn/xn to develop our integrand for any value of n will take us into the same territory.

We might consider trying to multiply our integrand by a2/a2, but it commutes with the integration operator and, thus, has no effect upon the integration. However, we might consider one of its isosums; to wit,

(Eq'n 4)

If we multiply our integrand by that formula, we get

(Eq'n 5)

If we integrate the second integral on the right side of that equation by parts with

(Eq'ns 6)

then we have

(Eq'n 7)

As with all integrals, we can prove and verify that result by differentiating it and comparing the differential with our original integrand.

Example Application:

Gravitational Deflection of Starlight-

The direct acceleration component

If we approximate the path of a pulse of light past the sun by a straight line with its perihelion a radial distance R from the sun's center, then the light accelerates in the direction perpendicular to the line at the rate

(Eq'n 8)

in which equation x represents the distance along the line from the perihelion point. Traveling along the line at the speed c, the pulse crosses the distance dx in the time interval dt = dx/c, so the pulse acquires the net velocity in the direction perpendicular to its motion along the line (the y-direction) of

(Eq'n 9)

To calculate the angle of deflection in radians we have

(Eq'n 10)

If we let x' tend toward infinity, then we approach the limit

(Eq'n 11)

Note that for our sun MG/c2 = 1.478 kilometers.


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