From [1,2]
2r^4 ln r dr
2r^4 ln r dr
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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
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
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∫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
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
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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
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
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use formula for integration of product of 2 functions
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2/25 (-31 + 32 Log[32])
6.39228
6.39228