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Question Number 175051 by Mathspace last updated on 17/Aug/22

$$findthevalueof$$$${\int }_0^{\mathrm{\infty }}\frac{arctanx}{{(x^2+1)}^3}dx$$

Commented by mokys last updated on 20/Aug/22

$$\mathit{I}={\int }_0^{\mathrm{\infty }}\frac{\mathit{a}\mathit{r}\mathit{c}\mathit{t}\mathit{a}\mathit{n}\mathit{x}}{{({\mathit{x}}^2+1)}^3}\mathit{d}\mathit{x}$$$$$$$$\mathit{x}=\mathit{t}\mathit{a}\mathit{n}\mathit{y}\to \mathit{d}\mathit{x}={\mathit{s}\mathit{e}\mathit{c}}^2\mathit{y}\mathit{d}\mathit{x}$$$$\mathit{x}=0\to \mathit{y}=0,\mathit{x}\to \mathrm{\infty }\Rightarrow \mathit{y}=\frac{\mathit{\pi }}{2}$$$$$$$$\mathit{I}={\int }_0^{\frac{\mathit{\pi }}{2}}\mathit{y}{\mathit{c}\mathit{o}\mathit{s}}^4\mathit{y}\mathit{d}\mathit{y}$$$$$$$$\mathit{u}=\mathit{y}\to \mathit{d}\mathit{u}=\mathit{d}\mathit{y},\mathit{d}\mathit{v}={\mathit{c}\mathit{o}\mathit{s}}^4\mathit{y}\to \mathit{v}=\frac{\mathit{s}\mathit{i}\mathit{n}4\mathit{y}+8\mathit{s}\mathit{i}\mathit{n}2\mathit{y}+12\mathit{y}}{32}$$$$$$$$\text{Missing left or extra right}$$$$$$$$\text{Missing left or extra right}$$$$$$$$\mathit{I}=\frac{3{\mathit{\pi }}^2}{32}-\frac{1}{32}[(-\frac{1}{4}+4+\frac{3{\mathit{\pi }}^2}{2})-(-\frac{1}{4}-4)]$$$$$$$$\mathit{I}=\frac{3{\mathit{\pi }}^2}{32}-\frac{1}{32}\left[\frac{16+3{\mathit{\pi }}^2}{2}\right]=\frac{6{\mathit{\pi }}^2-16-3{\mathit{\pi }}^2}{64}=\frac{3{\mathit{\pi }}^2-16}{64}$$$$$$$$AldolaimyMohammad$$$$$$

Commented by Tawa11 last updated on 20/Aug/22

$$\mathrm{Great}\mathrm{sir}.$$

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