Question

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Answer 1

$\int \frac{3x^2-2}{(x+1)(x^2+x+1)}dx=\int \frac{dx}{x+1}+\int \frac{2x-3}{x^2+x+1}dx=ln|x+1|+\int \frac{(2x-3)dx}{(x+\frac{1}{2})^2+\frac{3}{4}} =\\\\=\Big [\, t=x+\frac{1}{2}\; ,\; x=t-\frac{1}{2},\; dx=dt\, \Big ]=ln|x+1|+\int \frac{(2t-4)dt}{t^2+\frac{3}{4}}=\\\\=ln|x+1|+\int \frac{2t\, dt}{t^2+\frac{3}{4}}-4\int \frac{dt}{t^2+\frac{3}{4}}=\\\\=ln|x+1|+ln|t^2+\frac{3}{4}|-4\cdot \frac{2}{\sqrt3}\, arctg\frac{2t}{\sqrt3}+C=\\\\=ln|x+1)+ln|x^2+x+1|-\frac{8}{\sqrt3}\, arctg\frac{2x+1}{\sqrt3}+C\; .$