Find-out-A-n-0-0-pi-2-1-sinx-n-cosxdx-

Question Number 75080 by ~blr237~ last updated on 07/Dec/19

Find out

A = \underset{n = 0}{\sum^{\infty}} \int_{0}^{\frac{π}{2}} {(1 - \sqrt{sinx})}^{n} cosxdx

Commented by mathmax by abdo last updated on 07/Dec/19

w e h a v e ∣ 1 - \sqrt{s i n x} ∣< 1 \Rightarrow A = \int_{0}^{\frac{π}{2}} (\sum_{n = 0}^{\infty} {(1 - \sqrt{s i n x})}^{n}) c o s x d x

= \int_{0}^{\frac{π}{2}} \frac{1}{1 - (1 - \sqrt{s i n x})} c o s (x) d x = \int_{0}^{\frac{π}{2}} \frac{c o s x}{\sqrt{s i n x}} d x

c h a g e m e n t \sqrt{s i n x} = t g i v e s i n x = t^{2} \Rightarrow x = a r c s i n (t^{2}) \Rightarrow

d x = \frac{2 t}{\sqrt{1 - t^{4}}} \Rightarrow A = \int_{0}^{1} \frac{\sqrt{1 - t^{4}}}{t} \times \frac{2 t}{\sqrt{1 - t^{4}}} d t = 2 \int_{0}^{1} d t = 2