differentiating under the integral sign | differentiation under the integral sign

The Leibniz integral rule gives a formula for differentiation of a definite integral whose limits are functions of the differential variable, d/(dz) integral_(a(z))^(b(z)) f(x, z) d x = integral_(a(z))^(b(z)) (df)/(dz) d x + f(b(z), z)(db)/(dz) - f(a(z), z)(da)/(dz). It is sometimes known as differentiation under the integral sign. This rule can be used to evaluate certain unusual definite integrals such as ϕ(α) | = | integral_0^π ln(1 - 2α cos x + α^2) d x | = | 2π ln left bracketing bar α right bracketing bar for left bracketing bar α right bracketing bar >1.

derivative | integral | integration under the integral sign