find-1-1-e-x-1-x-2-dx-

Question Number 92134 by mathmax by abdo last updated on 05/May/20

f i n d \int_{- 1}^{1} \frac{e^{x}}{\sqrt{1 - x^{2}}} d x

Commented by mathmax by abdo last updated on 05/May/20

w e u s e t h e a p p r o x i m a t i o n \int_{- 1}^{1} \frac{f (x)}{\sqrt{1 - x^{2}}} d x \sim \frac{π}{n} \sum_{k = 1}^{n} f (c o s (\frac{(2 k - 1) π}{2 n}))

\Rightarrow \int_{- 1}^{1} \frac{e^{x}}{\sqrt{1 - x^{2}}} d x \sim \frac{π}{n} \sum_{k = 1}^{n} e^{c o s (\frac{(2 k - 1) π}{2 n})}

l e t t a k e n = 3 \Rightarrow \int_{- 1}^{1} \frac{e^{x}}{\sqrt{1 - x^{2}}} d x \sim \frac{π}{3} {e^{c o s (\frac{π}{6})} + e^{c o s (\frac{3 π}{6})} + e^{c o s (\frac{5 π}{6})}}

= \frac{π}{3} {e^{\frac{\sqrt{3}}{2}} + 1 + e^{- \frac{\sqrt{3}}{2}}} = \frac{π}{3} {2 c h (\frac{\sqrt{3}}{2}) + 1} \Rightarrow

\int_{- 1}^{1} \frac{e^{x}}{\sqrt{1 - x^{2}}} d x \sim \frac{π}{3} + \frac{2 π}{3} c h (\frac{\sqrt{3}}{2})