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Question Number 32714 by caravan msup abdo. last updated on 31/Mar/18

$$calculate{\int }_1^{+\mathrm{\infty }}\frac{dt}{t^2\sqrt{1+t^2}}.$$

Commented by abdo imad last updated on 03/Apr/18

$$ch.t=tan\theta giveI={\int }_{\frac{\pi }{4}}^{\frac{\pi }{2}}\frac{1}{{tan}^2\theta .}cos\theta .(1+{tan}^2\theta )d\theta$$$$={\int }_{\frac{\pi }{4}}^{\frac{\pi }{2}}\frac{d\theta }{cos\theta {tan}^2\theta }={\int }_{\frac{\pi }{4}}^{\frac{\pi }{2}}\frac{cos\theta }{{sin}^2\theta }d\theta ={[-\frac{1}{sin\theta }]}_{\frac{\pi }{4}}^{\frac{\pi }{2}}$$$$=\frac{1}{\frac{\sqrt{2}}{2}}-1=\frac{2}{\sqrt{2}}-1=\sqrt{2}-1.$$

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