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In integration by parts (∫u dv = uv − ∫v du), the LIATE rule helps choose 'u'. For ∫x ln(x) dx, the correct choice of 'u' is:
A$u = x$ (the algebraic part of the integrand)
B$u = \ln(x)$ (logarithmic, higher in LIATE)
C$u = x \ln(x)$ (the entire integrand)
D$u = e^x$ (exponential trick)
Answer & Solution
Correct answer: B. $u = \ln(x)$ (logarithmic, higher in LIATE)
LIATE: L (logarithmic) is highest priority. So u = ln(x), dv = x dx. Result: (x²/2) ln(x) − ∫(x²/2)(1/x) dx = (x²/2) ln(x) − x²/4 + C.
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