integral of (ln x)^2/x (u substitution)

[@gelbkehlchen]

Solution: ∫ln²(x)/x*dx = ------------------------------------ 1st solution by partial integration: Partial integration can be derived from the product rule of differential calculus. The product rule of differential calculus states: (u*v)’ = u’*v+u*v’ |-u’*v ⟹ (u*v)’-u’*v = u*v’ ⟹ u*v’ = (u*v)’-u’*v |∫() ⟹ ∫u*v’*dx = u*v-∫u’*v*dx ------------------------------------ ∫ln²(x)*1/x*dx = ln³(x)-∫2*ln(x)*1/x*ln(x)*dx = ln³(x)-2*∫ln²(x)*1/x*dx |+2*∫ln²(x)*1/x*dx ⟹ 3*∫ln²(x)*1/x*dx = ln³(x)+C |/3 ⟹ ∫ln²(x)*1/x*dx = ∫ln²(x)/x*dx = ln³(x)/3+C/3 ------------------------------ 2nd solution by substitution: u = ln(x) du = 1/x*dx ------------------------------ = ∫u²*du = u³/3+C = ln³(x)/3+C Checking the result by deriving: [ln³(x)/3+C]’ = ln²(x)/x everything okay!

[@AnanyaSinthyew]

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