Defining v[u] = -Integrate[f[u], u] the field then has Lagrangian density (( ∂ t u) 2 - ( ∂ x u) 2 )/2 - v[u] and conserves the Hamiltonian (energy function) Integrate[(( ∂ t u) 2 + ( ∂ t u) 2 )/2 + v[u], {x, - ∞ , ∞ }] With the choice for f[u] made here (with a ≥ 0 ), v[u] is bounded from below, and as a result it follows that no singularities ever occur in u[t, x] .

Frameworks [in mathematics] Symbolic integration was in the past done by a collection of ad hoc methods like substitution, partial fractions, integration by parts, and parametric differentiation. But in Mathematica Integrate is now almost completely systematic, being based on structure theorems for finding general forms of integrals, and on general representations in terms of MeijerG and other functions.

[Conserved quantities in] PDEs In the early 1960s it was discovered that certain nonlinear PDEs support an infinite number of distinct conserved quantities, associated with so-called integrability and the presence of solitons.

An example is Monte Carlo integration, where what ultimately matters is uniform sampling of the integrand—which can usually be achieved better by quasi-random irrational number multiple (see page 903 ) or digit reversal (see page 905 ) sequences than by sequences one might consider more random.

Here a typical orthogonality property is Integrate[f[r, x] f[s, x], {x, 0, 1}]  KroneckerDelta[r, s] .

Mathematical impossibilities It is sometimes said that in the 1800s problems such as trisecting angles, squaring the circle, solving quintics, and integrating functions like Exp[x 2 ] were proved mathematically impossible.

For rational functions f[x] , Integrate[f[x], {x, 0, 1}] must always be a linear function of Log and ArcTan applied to algebraic numbers ( f[x] = 1/(1 + x 2 ) for example yields π /4 ). Multiple integrals of rational functions can be more complicated, as in Integrate[1/(1 + x 2 + y 2 ), {x, 0, 1}, {y, 0, 1}]  HypergeometricPFQ[{1/2, 1, 1}, {3/2, 3/2}, 1/9]/6 + 1/2 π ArcSinh[1] - Catalan and presumably often cannot be expressed at all in terms of standard mathematical functions.

If one evaluates NIntegrate or NDSolve by effectively fitting functions to order s polynomials the difficulty of getting results with n -digit precision typically increases like 2 n/s . An adaptive algorithm such as Romberg integration reduces this to about 2^ √ n .

Such a circle has area 2 π a 2 (1 - Cos[r/a]) = π r 2 (1 - r 2 /(12 a 2 ) + r 4 /(360a 4 ) - …) In the d -dimensional space corresponding to the surface of a (d + 1) -dimensional sphere of radius a , the volume of a d -dimensional sphere of radius r is similarly given by d s[d] a d Integrate[Sin[ θ ] d - 1 , { θ ,0, r/a}] = s[d] r d (1 - d (d - 1) r 2 /((6 (d + 2))a 2 + (d (5d 2 - 12d + 7))r 4 /((360 (d + 4))a 4 ) …) where Integrate[Sin[x] d - 1 , x] = -Cos[x] Hypergeometric2F1[1/2, (2 - d)/2, 3/2, Cos[x] 2 ] In an arbitrary d -dimensional space the volume of a sphere can depend on position, but in general it is given by s[d] r d (1 - RicciScalar r 2 /(6(d + 2)) + …) where the Ricci scalar curvature is evaluated at the position of the sphere.

A somewhat better approximation is LogIntegral[n] , equal to Integrate[1/Log[t], {t, 2, n}] .