Integration under the integral sign is the use of the identity integral_a^b d x integral_(α_0)^α f(x, α) d α = integral_(α_0)^α d α integral_a^b f(x, α) d x to compute an integral. For example, consider integral_0^1 x^α d x = 1/(α + 1) for α>-1. Multiplying by d α and integrating between a and b gives integral_a^b d α integral_0^1 x^α d x | = | integral_a^b (d α)/(α + 1) | = | ln left bracketing bar (b + 1)/(a + 1) right bracketing bar .

integral | integral sign | integration | Leibniz integral rule