$I=\int\limits^1_0\frac{x^3+1-1}{x+1}dx=\int\limits^1_0(x^2-x+1-\frac{1}{x+1})dx$$=(\frac{x^3}{3}-\frac{x^2}{2}+x-ln|x+1|)|^1_0=\frac{5}{6}-ln 2$
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