Интегрирование по частям
u = lnx
du = (lnx) ' = dx/x
dv = x^3 dx
v = ∫ x^3 dx = x^4/4
u v - ∫ v du =
= lnx x^4/4 - 1/4 ∫ x^4/x dx =
= lnx *x^4/4 - 1/4 ∫ x^3 dx =
= lnx * x^4/4 - 1/4 * x^4/4 =
= lnx * x^4/4 - x^4/16 =
= x^4/4 * ( lnx - 1/4)
По формуле Ньютона-Лейбница получим
e^4/4 * ( lne - 1/4) - 1/4 * ( ln1 - 1/4) =
= e^4/4 ( 4/4 - 1/4) - 1/4 (0 - 1/4)=
= 3e^4/16 + 1/16 =
= 1/16 * (3e^4 + 1)