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Question Number 159592 by mnjuly1970 last updated on 19/Nov/21

$$$$$$calculate:$$$$$$$$\mathcal{I}:={\int }_0^{\mathrm{\infty }}{\left(\frac{arctan\left(x\right)}{x}\right)}^{3}dx=?$$$$$$

Answered by mindispower last updated on 19/Nov/21

$$byart=\frac{3}{2}{\int }_0^{\mathrm{\infty }}\frac{{arctan}^2\left(x\right)}{x^2(1+x^2)}dx$$$$=3{\int }_0^{\mathrm{\infty }}\frac{arctan\left(x\right)}{x(1+x^2)}dx-\frac{1}{2}.{\left(\frac{\pi }{2}\right)}^3$$$$=3{\int }_0^{\frac{\pi }{2}}\frac{xcos\left(x\right)}{sin\left(x\right)}-\frac{{\pi }^3}{16}$$$$-3{\int }_0^{\frac{\pi }{2}}ln\left(sin\left(x\right)\right)dx-\frac{{\pi }^3}{16}$$$$\frac{3\pi }{2}ln\left(2\right)-\frac{{\pi }^3}{16}$$

Commented by mnjuly1970 last updated on 19/Nov/21

$$thanksalotsirpower$$

Commented by mindispower last updated on 19/Nov/21

$$withepleasursir$$

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