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Question Number 200403 by Anonim_X last updated on 18/Nov/23

$$$$$$\int \frac{({\mathit{x}}^2+1)\mathit{d}\mathit{x}}{\mathit{x}(\mathit{x}-1)(\mathit{x}+1)}=??$$$$$$

Answered by cortano12 last updated on 18/Nov/23

$$\mathrm{I}=\int \frac{{\mathrm{x}}^2-1+2}{\mathrm{x}(\mathrm{x}-1)(\mathrm{x}+1)}\mathrm{dx}$$$$\mathrm{I}=\ln \mid \mathrm{x}\mid +2\int \frac{\mathrm{dx}}{\mathrm{x}(\mathrm{x}+1)(\mathrm{x}-1)}$$$$\mathrm{I}=\ln \mid \mathrm{x}\mid +2\int (-\frac{1}{\mathrm{x}}+\frac{1}{2(\mathrm{x}+1)}+\frac{1}{2(\mathrm{x}-1)})\mathrm{dx}$$$$\mathrm{I}=-\ln \mid \mathrm{x}\mid +\ln \mid \mathrm{x}+1\mid +\ln \mid \mathrm{x}-1\mid +\mathrm{c}$$

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