Evaluate the intergal

From [1,2]

2r^4 ln r dr

First we integrate by parts:

u = ln(r) . . . . dv = 2r⁴ dr
du = 1/r dr . . . v = 2/5 r⁵

∫ u dv = u v − ∫ v du

∫ 2r⁴ ln(r) dr = 2/5 r⁵ ln(r) − ∫ 2/5 r⁴ dr
. . . . . . . . . .= 2/5 r⁵ ln(r) − 2/25 r⁵

Evaluating from 1 to 2 we get

∫₁² 2r⁴ ln(r) dr = (2/5 r⁵ ln(r) − 2/25 r⁵) |₁²
. . . . . . . . . . . = 2/5 (32) ln(2) − 2/25 (32) − 2/5 ln(1) + 2/25
. . . . . . . . . . . = 64/5 ln(2) − 62/25
. . . . . . . . . . . ≈ 6.392283911

∫2r^4*ln(r) dr from 1 to 2

u = ln(r)

dv = 2r^4 dr

v = (2r^5)/5

du = dr/r

uv - ∫v du

(2r^5)/5*ln(r) - 2/5*∫r^4 dr

(2r^5)/5*ln(r) - 2r^5/25 eval. from 1 to 2

= 64/5*ln(2) - 62/25

use integration by parts.

u=lnr dv=2r^4
du=1/r v=(2/5)r^5

(2/5)r^5*(lnr)-2/5*INT[r^4dr]

(2/5)r^5*lnr-(2/25)*r^5.
plug in 2 then plug in 1

use formula for integration of product of 2 functions

2/25 (-31 + 32 Log[32])

6.39228