Question

Sa se calculeze l =∫ x^3 ln ⁡xdx

Answer 1

Voi folosi metoda integrarii prin parti cu formula

∫f×g` dx = f×g - ∫f`×g dx

I = ∫ x³ × lnx dx = ∫ lnx × x³

f(x) = lnx => f`(x) = 1/x

g`(x) = x³ => g(x) = x⁴/4

I = lnx×x³ - ∫ 1/x×x⁴/4 dx =

lnx×x³ - ∫ x³/4 dx =

lnx×x³ - 1/4∫ x³ dx =

lnx×x³ - 1/4×x⁴/4 + C =

lnx×x³ - x⁴/16 + C