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Question Number 87086 by Chi Mes Try last updated on 02/Apr/20

Answered by mind is power last updated on 03/Apr/20

$$={\int }_0^{+\mathrm{\infty }}2\frac{dt}{(t^4+(1+2\sqrt{2})t^2+1)(1-t^2+t^4-.........+t^{100})}=A$$$$=2{\int }_0^{+\mathrm{\infty }}\frac{dt}{t^{104}(1+\frac{1+2\sqrt{2}}{t^2}+\frac{1}{t^4})(\frac{1}{t^{100}}-\frac{1}{t^{98}}+.......+1)}$$$$\frac{1}{t}=y\Rightarrow$$$$2{\int }_0^{+\mathrm{\infty }}\frac{y^{102}}{(1+(1+2\sqrt{2})y^2+y^4)(1-y^2+.........+y^{100})}$$$$1-y^2+.......+y^{100}=\frac{1-{(-y^2)}^{51}}{1+y^2}=\frac{1+y^{102}}{1+y^2}$$$$2A={\int }_0^{+\mathrm{\infty }}\frac{2+2x^{102}dx}{(1+(1+2\sqrt{2})x^2+x^4)(1-x^2+x^4-.....+x^{100})}$$$$\Rightarrow A={\int }_0^{+\mathrm{\infty }}\frac{1}{(1+(1+2\sqrt{2})x^2+x^4)}.\frac{1+x^{102}}{(1-x^2+........+x^{100})}dx$$$$A={\int }_0^{+\mathrm{\infty }}\frac{dx}{(1+(1+2\sqrt{2})x^2+x^4)}$$$$easynow$$$$$$

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