Calculus II · Integration techniques and improper integrals
Expose a rational remainder after integration by parts
Problem
Evaluate exactly $$ I=\int_0^1 x^2\ln(1+x)\,dx. $$ Choose the parts explicitly, retain the endpoint term, and divide the polynomial in the remaining fraction before integrating. Simplify the answer to a linear combination of $\ln2$ and a rational number. Finally, use $0\leq\ln(1+x)\leq\ln2$ on $[0,1]$ to check the sign and size of your result.
Hint
Take u=ln(1+x) and dv=x^2 dx; first compute the boundary contribution at both 0 and 1.
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