Скачать или смотреть how to integrate (ln x)^3✨use integration by parts repeatedly

integral of (ln x)^3 dx
how to integrate (ln x)^3
use integration by parts repeatedly
let u = (ln x)^3, dv = dx, du = 3 (ln x)^2 * (1/x) dx, v = x
∫ (ln x)^3 dx = x (ln x)^3 - ∫ x * 3 (ln x)^2 * (1/x) dx = x (ln x)^3 - 3 ∫ (ln x)^2 dx
now need ∫ (ln x)^2 dx, which by parts: = x (ln x)^2 - 2x ln x + 2x (from earlier)
thus ∫ (ln x)^3 dx = x (ln x)^3 - 3 [ x (ln x)^2 - 2x ln x + 2x ] + C
= x (ln x)^3 - 3x (ln x)^2 + 6x ln x - 6x + C
step-by-step integration by parts
technique: reduce power of ln x step by step
antiderivative: x (ln x)^3 - 3x (ln x)^2 + 6x ln x - 6x + C
example of integrating higher power of ln x
detailed solution with recurrence
calculus integral with (ln x)^3
method: integration by parts formula
verify by differentiating result
pattern: ∫ (ln x)^n dx = x (ln x)^n - n ∫ (ln x)^(n-1) dx
use this recurrence
final answer expanded
step-by-step derivation
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