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Question Number 156820 by tabata last updated on 15/Oct/21

$$\int \frac{dx}{(x^2-x+1)\left(\sqrt{x^2+x+1}\right)}$$

Commented by tabata last updated on 15/Oct/21

$$?????$$

Commented by MJS_new last updated on 16/Oct/21

$$\mathrm{is}\mathrm{this}\mathrm{the}\mathrm{Annual}\mathrm{Question}\mathrm{Mark}\mathrm{Convention}?$$$$?????????????????????????????????????$$

Commented by aliyn last updated on 16/Oct/21

$$???????$$

Answered by MJS_new last updated on 16/Oct/21

$$\int \frac{dx}{(x^2-x+1)\sqrt{x^2+x+1}}=$$$$[t=\frac{\sqrt{3}{t}^2-2t-\sqrt{3}}{4t}\to dx=\frac{\sqrt{3(x^2+x+1)}}{2x+1+2\sqrt{x^2+x+1}}]$$$$=\frac{16}{3}\int \frac{t}{t^4-\frac{8\sqrt{3}}{3}{t}^3+\frac{22}{3}{t}^2+\frac{8\sqrt{3}}{3}t+1}dt=$$$$=\frac{16}{3}\int \frac{t}{(t^2-(2\sqrt{2}-\frac{4\sqrt{3}}{3})t+5-2\sqrt{6})(t^2-(2\sqrt{2}+\frac{4\sqrt{3}}{3})t+5+2\sqrt{6})}dt$$$$\mathrm{now}\mathrm{do}\mathrm{the}\mathrm{decomposition}\mathrm{yourself}!!!!!!!!$$

Commented by tabata last updated on 16/Oct/21

$$\mathit{i}\mathit{t}\mathit{s}\mathit{v}\mathit{e}\mathit{r}\mathit{y}\mathit{h}\mathit{a}\mathit{r}\mathit{d}\mathit{q}\mathit{u}\mathit{e}\mathit{s}\mathit{t}\mathit{i}\mathit{o}\mathit{n}$$

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