calculate-1-dt-t-2-1-t-2-

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

c a l c u l a t e \int_{1}^{+ \infty} \frac{d t}{t^{2} \sqrt{1 + t^{2}}} .

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

c h . t = t a n θ g i v e I = \int_{\frac{π}{4}}^{\frac{π}{2}} \frac{1}{{t a n}^{2} θ .} c o s θ . (1 + {t a n}^{2} θ) d θ

= \int_{\frac{π}{4}}^{\frac{π}{2}} \frac{d θ}{c o s θ {t a n}^{2} θ} = \int_{\frac{π}{4}}^{\frac{π}{2}} \frac{c o s θ}{{s i n}^{2} θ} d θ = {[- \frac{1}{s i n θ}]}_{\frac{π}{4}}^{\frac{π}{2}}

= \frac{1}{\frac{\sqrt{2}}{2}} - 1 = \frac{2}{\sqrt{2}} - 1 = \sqrt{2} - 1 .