Binomial Differential

binomial differential

[bī′nō·mē·əl ‚dif·ə′ren·chəl]
A differential of the form xp (a + bxq) rdx, where p, q, r are integers.

Binomial Differential


an expression of the form

xm (a + bxn)pdx

where a and b are nonzero constants and m, n, and p are rational numbers. The integral of a binomial differential ∫xm (a + bxn)p dx is expressed in finite form through elementary functions only in three cases: (1) if p is an integer; (2) if [(m + 1)/n ] + p is an integer; or (3) if (m + l)/n is an integer. These three conditions for the integrability of binomial differentials were known to L. Euler. P. L. Chebyshev demonstrated in 1853 that in all other cases the integral of a binomial differential is not expressed in finite form through elementary functions. This was one of the first cases where the question of whether integrability in finite form is possible was resolved for a sufficiently general class of analytic expressions. Chebyshev’s result ranks alongside other classical theorems on the impossibility of algebraic solutions for various classes of algebraic equations and on the unsolvability of the problem of squaring a circle using a straightedge and compass.