GATE 2022 MN – Question 62
Evaluate $\int_0^1 x\log(1+x)\,dx$, using the natural logarithm. Give two decimals.
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Show answer and explanation
Correct answer: 0.25 to 0.25
Explanation
Integration by parts with u=ln(1+x), dv=x dx gives $I=\frac12\ln2-\frac12\int_0^1x^2/(1+x)\,dx$. Polynomial division gives x²/(1+x)=x−1+1/(1+x), whose integral over[0, 1] is ln 2−1/2. Hence I=(ln 2)/2−(ln 2−1/2)/2=1/4=0.25.