A product of 2 variables under the integral sign can be obtained a little more easily, because of the curious powers of the differential analyzer. Thus, if it is desired to obtain ∫ xy dt, we can use the formula:
| xy dt = | x d | y dt |
and this operation does not require an adder. The connections are as follows:
| Shaft t | To Integrator 1, Disc |
| Shaft y | To Integrator 1, Screw |
| Integrator 1, Wheel | To Integrator 2, Disc |
| Shaft x | To Integrator 2, Screw |
| Integrator 2, Wheel | To Shaft expressing ∫xy dt |
In order to get the quotient of 2 variables, x/y, we can use some more tricks. First, the reciprocal 1/y can be obtained by using the two simultaneous equations:
 | 1 | dy = log y, | |
| y | |||
| - | 1 | d(log y) = y |
| y |
The connections are as follows:
| Shaft y | To Integrator 1, Disc and to Integrator 2, Wheel |
| Shaft log y | To Integrator 1, Wheel and to Integrator 2, Disc |
| Shaft 1/y | To Integrator 1, Screw, and negatively to Integrator 2, Screw |
In order to get x/y, we can then multiply x by 1/y. We see that this setup gives us log y for nothing, that is, without needing more integrators or other equipment. Clearly, other tricks like this will give sin x, cos x, eˣ, x², and other functions that satisfy simple differential equations.
An integral of a reciprocal can be obtained even more directly. Suppose that