find-I-0-1-1-t-1-t-dt-

Question Number 30512 by abdo imad last updated on 22/Feb/18

f i n d I = \int_{0}^{1} \sqrt{\frac{1 - t}{1 + t}} d t .

Commented by abdo imad last updated on 24/Feb/18

t h e c h . t = c o s θ g i v e I = \int_{0}^{\frac{π}{2}} \sqrt{\frac{1 - c o s θ}{1 + c o s θ}} s i n θ d θ

= \int_{0}^{\frac{π}{2}} t a n (\frac{θ}{2}) s i n θ d . θ = \int_{0}^{\frac{π}{2}} \frac{s i n (\frac{θ}{2})}{c o s (\frac{θ}{2})} 2 s i n (\frac{θ}{2}) c o s (\frac{θ}{2}) d θ

= \int_{0}^{\frac{π}{2}} 2 {s i n}^{2} (\frac{θ}{2}) d θ = \int_{0}^{\frac{π}{2}} (1 - c o s θ) d θ = \frac{π}{2} - {[s i n θ]}_{0}^{\frac{π}{2}}

= \frac{π}{2} - 1 .