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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

=∫_0 ^(+∞) 2(dt/((t^4 +(1+2(√2))t^2 +1)(1−t^2 +t^4 −.........+t^(100) )))=A =2∫_0 ^(+∞) (dt/(t^(104) (1+((1+2(√2))/t^2 )+(1/t^4 ))((1/t^(100) )−(1/t^(98) )+.......+1))) (1/t)=y⇒ 2∫_0 ^(+∞) (y^(102) /((1+(1+2(√2))y^2 +y^4 )(1−y^2 +.........+y^(100) ))) 1−y^2 +.......+y^(100) =((1−(−y^2 )^(51) )/(1+y^2 ))=((1+y^(102) )/(1+y^2 )) 2A=∫_0 ^(+∞) ((2+2x^(102) dx)/((1+(1+2(√2))x^2 +x^4 )(1−x^2 +x^4 −.....+x^(100) ))) ⇒A=∫_0 ^(+∞) (1/((1+(1+2(√2))x^2 +x^4 ))).((1+x^(102) )/((1−x^2 +........+x^(100) )))dx A=∫_0 ^(+∞) (dx/((1+(1+2(√2))x^2 +x^4 ))) easy now

$$=\int_{\mathrm{0}} ^{+\infty} \mathrm{2}\frac{{dt}}{\left({t}^{\mathrm{4}} +\left(\mathrm{1}+\mathrm{2}\sqrt{\mathrm{2}}\right){t}^{\mathrm{2}} +\mathrm{1}\right)\left(\mathrm{1}−{t}^{\mathrm{2}} +{t}^{\mathrm{4}} −.........+{t}^{\mathrm{100}} \right)}={A} \\ $$$$=\mathrm{2}\int_{\mathrm{0}} ^{+\infty} \frac{{dt}}{{t}^{\mathrm{104}} \left(\mathrm{1}+\frac{\mathrm{1}+\mathrm{2}\sqrt{\mathrm{2}}}{{t}^{\mathrm{2}} }+\frac{\mathrm{1}}{{t}^{\mathrm{4}} }\right)\left(\frac{\mathrm{1}}{{t}^{\mathrm{100}} }−\frac{\mathrm{1}}{{t}^{\mathrm{98}} }+.......+\mathrm{1}\right)} \\ $$$$\frac{\mathrm{1}}{{t}}={y}\Rightarrow \\ $$$$\mathrm{2}\int_{\mathrm{0}} ^{+\infty} \frac{{y}^{\mathrm{102}} }{\left(\mathrm{1}+\left(\mathrm{1}+\mathrm{2}\sqrt{\mathrm{2}}\right){y}^{\mathrm{2}} +{y}^{\mathrm{4}} \right)\left(\mathrm{1}−{y}^{\mathrm{2}} +.........+{y}^{\mathrm{100}} \right)} \\ $$$$\mathrm{1}−{y}^{\mathrm{2}} +.......+{y}^{\mathrm{100}} =\frac{\mathrm{1}−\left(−{y}^{\mathrm{2}} \right)^{\mathrm{51}} }{\mathrm{1}+{y}^{\mathrm{2}} }=\frac{\mathrm{1}+{y}^{\mathrm{102}} }{\mathrm{1}+{y}^{\mathrm{2}} } \\ $$$$\mathrm{2}{A}=\int_{\mathrm{0}} ^{+\infty} \frac{\mathrm{2}+\mathrm{2}{x}^{\mathrm{102}} {dx}}{\left(\mathrm{1}+\left(\mathrm{1}+\mathrm{2}\sqrt{\mathrm{2}}\right){x}^{\mathrm{2}} +{x}^{\mathrm{4}} \right)\left(\mathrm{1}−{x}^{\mathrm{2}} +{x}^{\mathrm{4}} −.....+{x}^{\mathrm{100}} \right)} \\ $$$$\Rightarrow{A}=\int_{\mathrm{0}} ^{+\infty} \frac{\mathrm{1}}{\left(\mathrm{1}+\left(\mathrm{1}+\mathrm{2}\sqrt{\mathrm{2}}\right){x}^{\mathrm{2}} +{x}^{\mathrm{4}} \right)}.\frac{\mathrm{1}+{x}^{\mathrm{102}} }{\left(\mathrm{1}−{x}^{\mathrm{2}} +........+{x}^{\mathrm{100}} \right)}{dx} \\ $$$${A}=\int_{\mathrm{0}} ^{+\infty} \frac{{dx}}{\left(\mathrm{1}+\left(\mathrm{1}+\mathrm{2}\sqrt{\mathrm{2}}\right){x}^{\mathrm{2}} +{x}^{\mathrm{4}} \right)}\:\: \\ $$$${easy}\:{now} \\ $$$$ \\ $$

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