An integral of a square wave is a spline approximation of a sine wave. The third integral is a cubic-spline approximation, which has very low harmonic distortion. The amplitude of the approximated sine wave is the amplitude of the square wave scaled by the Nth power of the period; for a third-order spline, it’s scaled by the cube of the period.
This is interesting because the discrete analog of integral, running sum, is linear, time-invariant, and very cheap indeed to compute (one accumulator and one addition per sample), and sine waves are useful primitives for composing many signals.
In the case of audio synthesis, in particular, I’m thinking you can take some square waves (or envelope-shaped square waves), add them together, and take their running sum a few times, to get a mix of sinusoidal signals.
To be concrete, suppose you’re synthesizing 16ksps audio, and you want to be able to cover the audible range down to 20Hz and up to the Nyquist frequency at 8kHz. A 20Hz square wave has a period of 16000/40 = 400 samples on both the top and bottom; this results in amplifying the original square wave by about 400³ for a third-order spline, or 64 million. (This is not quite correct because of discretization; the actual number is 10.6 to 10.7 million.) This means you need 32-bit integer math for your accumulators (and for the amplitudes of your higher frequencies), but that 32-bit math gives you a dynamic range at 20Hz of 10dB log₁₀((2³¹/10.6M)²) = 46dB, which is quite respectable.
At higher frequencies, you have correspondingly more dynamic range; at Nyquist, you have 31 bits of dynamic range, since the square wave is the sine wave, or 187 dB.
So you have square waves, which are more or less cheap to compute (at least as long as you’re far from Nyquist and can therefore disregard jitter), and which you sum; and then you integrate them (three times) to get a sum of (cubic) spline approximations of sine waves. You have to scale the amplitudes of the square waves as mentioned above, proportional to the Nth power of the frequency.