"The world's sneakiest substitution." — Michael Spivak
\[\int R(\sin\omega, \cos\omega)\,d\omega\]
is a special case of Transform 5 in the USM framework, corresponding to the circular case with parameters \(a = 1\) and \(b = 0\).
Derivation
Let
\[I = \int R(\sin\omega, \cos\omega)\,d\omega.\]
Set \(x = \sin\omega\), so that
\[\cos\omega = \sqrt{1 - x^2} \quad (\text{using the principal square root, e.g., } \cos\omega \geq 0 \text{ for } \omega \in [-\pi/2,\pi/2]),\]
and
\[d\omega = \frac{dx}{\cos\omega} = \frac{dx}{\sqrt{1 - x^2}}.\]
Hence,
\[I = \int \frac{R\!\left(x, \sqrt{1 - x^2}\right)}{\sqrt{1 - x^2}}\,dx.\]
Transform 5 handles integrals involving \(\sqrt{a^2 - (x+b)^2}\) on the domain \(|y| \leq 1\) with \(y = (x+b)/a\). Take \(a = 1\), \(b = 0\) (so \(y = x\)) and let the parameter be \(r\) (as in the paper). The transform gives:
\[x = \frac{2r}{1+r^2}, \quad \sqrt{1 - x^2} = \frac{1 - r^2}{1 + r^2}, \quad dx = \frac{2(1 - r^2)}{(1 + r^2)^2}\,dr.\]
Substitute into the integral
\[\begin{aligned} I &= \int \frac{R\!\left(x, \sqrt{1 - x^2}\right)}{\sqrt{1 - x^2}}\,dx \\ &= \int \frac{R\!\left(\frac{2r}{1+r^2}, \frac{1 - r^2}{1 + r^2}\right)}{\frac{1 - r^2}{1 + r^2}} \cdot \frac{2(1 - r^2)}{(1 + r^2)^2}\,dr \\ &= \int R\!\left(\frac{2r}{1+r^2}, \frac{1 - r^2}{1 + r^2}\right)\frac{2}{1 + r^2}\,dr.\end{aligned}\]
The final expression is exactly the Weierstrass substitution formula:
\[\int R(\sin\omega, \cos\omega)\,d\omega= \int R\!\left(\frac{2r}{1+r^2}, \frac{1 - r^2}{1 + r^2}\right) \frac{2}{1 + r^2}\,dr.\]
On the principal branch where \(\psi=\sin^{-1}(x)=\omega\), this parameter is
\[r=\tan\!\Bigl(\frac{\psi}{2}\Bigr)=\tan\!\Bigl(\frac{\omega}{2}\Bigr),\]
Thus, the Weierstrass substitution emerges naturally from Transform 5 by setting \(a = 1\), \(b = 0\) and interpreting the integrand appropriately. This demonstrates (again!) that the USM unifies and generalizes classical substitution techniques (refer to Section 6 in the paper, which details how the USM also encompasses Euler substitutions 1 and 2), including the half‑angle tangent substitution of Weierstrass. Moreover, USM creates new "Weierstrass-like" substitutions (Transforms 1 & 2) that work for hyperbolic/algebraic regions ($|y| \ge 1$) where the standard $\tan(\omega/2)$ is not typically applied.
