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Question Number 186352 by normans last updated on 03/Feb/23

$$$$$${\int }_1^2\frac{{\mathit{t}\mathit{a}\mathit{n}}^{-1}\left(\mathit{x}\right)+2}{{\mathit{x}}^2}\mathit{d}\mathit{x}$$$$$$

Answered by MJS_new last updated on 04/Feb/23

$$\int \frac{2+\arctan x}{x^2}dx=2\int \frac{dx}{x^2}+\int \frac{\arctan x}{x^2}dx$$$$2\int \frac{dx}{x^2}=-\frac{2}{x}+C_1$$$$\int \frac{\arctan x}{x^2}dx=-\frac{\arctan x}{x}+\int \frac{dx}{x(x^2+1)}$$$$\int \frac{dx}{x(x^2+1)}=\int (\frac{1}{x}-\frac{x}{x^2+1})dx=\frac{1}{2}\ln \frac{x^2}{x^2+1}+C_2$$$$\Rightarrow$$$$\mathrm{answer}\mathrm{is}1-\frac{\ln 5-3\ln 2}{2}+\frac{1}{2}\arctan \frac{1}{2}$$

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