calculate-0-1-dt-t-1-t-2-

Question Number 31504 by abdo imad last updated on 09/Mar/18

c a l c u l a t e \int_{0}^{1} \frac{d t}{t + \sqrt{1 - t^{2}}} .

Commented by abdo imad last updated on 15/Mar/18

l e t p u t t = s i n x \Rightarrow I = \int_{0}^{\frac{π}{2}} \frac{c o s x d x}{s i n x + c o s x}

= \int_{0}^{\frac{π}{2}} \frac{c o s x + s i n x - s i n x}{s i n x + c o s x} d x = \frac{π}{2} - \int_{0}^{\frac{π}{2}} \frac{s i n x}{s i n x + c s x} d x

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

= I \Rightarrow 2 I = \frac{π}{2} \Rightarrow I = \frac{π}{4} .