find ∫_(−1) ^1 (e^x /(√(1−x^2 )))dx


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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
$we use the approximation ∫_(−1) ^1 ((f(x))/(√(1−x^2 )))dx ∼(π/n)Σ_(k=1) ^n f(cos((((2k−1)π)/(2n)))) ⇒∫_(−1) ^1 (e^x /(√(1−x^2 )))dx ∼(π/n)Σ_(k=1) ^n e^(cos((((2k−1)π)/(2n)))) let take n=3 ⇒∫_(−1) ^1 (e^x /(√(1−x^2 )))dx ∼(π/3){ e^(cos((π/6))) +e^(cos(((3π)/6))) +e^(cos(((5π)/6))) } =(π/3){ e^((√3)/2) + 1 + e^(−((√3)/2)) } =(π/3){2ch(((√3)/2))+1} ⇒ ∫_(−1) ^1 (e^x /(√(1−x^2 )))dx ∼ (π/3) +((2π)/3)ch(((√3)/2))$
$w e u s e t h e a p p r o x i m a t i o n \int_{- 1}^{1} \frac{f (x)}{\sqrt{1 - x^{2}}} d x \sim \frac{π}{n} \sum_{k = 1}^{n} f (c o s (\frac{(2 k - 1) π}{2 n}))$ $\Rightarrow \int_{- 1}^{1} \frac{e^{x}}{\sqrt{1 - x^{2}}} d x \sim \frac{π}{n} \sum_{k = 1}^{n} e^{c o s (\frac{(2 k - 1) π}{2 n})}$ $l e t t a k e n = 3 \Rightarrow \int_{- 1}^{1} \frac{e^{x}}{\sqrt{1 - x^{2}}} d x \sim \frac{π}{3} {e^{c o s (\frac{π}{6})} + e^{c o s (\frac{3 π}{6})} + e^{c o s (\frac{5 π}{6})}}$ $= \frac{π}{3} {e^{\frac{\sqrt{3}}{2}} + 1 + e^{- \frac{\sqrt{3}}{2}}} = \frac{π}{3} {2 c h (\frac{\sqrt{3}}{2}) + 1} \Rightarrow$ $\int_{- 1}^{1} \frac{e^{x}}{\sqrt{1 - x^{2}}} d x \sim \frac{π}{3} + \frac{2 π}{3} c h (\frac{\sqrt{3}}{2})$
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