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Question Number 40619 by math khazana by abdo last updated on 25/Jul/18

l e t f (x) = \int_{0}^{\frac{π}{2}} \frac{d θ}{x + {c o s}^{2} θ} w i t h x > 0 .

1) c a l c u l a t e f (x) a n d f^{^{'}} (x)

2) f i n d f^{(n)} (x) a n d f^{(n)} (0)

3) d e v e l o p p f a t i n t e g r s e r i e .

Commented by abdo mathsup 649 cc last updated on 27/Jul/18

1) w e h a v e p r o v e d t h a t f (x) = \frac{π \sqrt{2}}{2 \sqrt{x}} \Rightarrow

f^{^{'}} (x) = \frac{π \sqrt{2}}{2} {(x^{- \frac{1}{2}})}^{^{'}} = - \frac{1}{2} \frac{π \sqrt{2}}{2} x^{- \frac{3}{2}} = - \frac{π \sqrt{2}}{4 x \sqrt{x}}

2) f^{(n)} (x) = \frac{π \sqrt{2}}{2} {(x^{- \frac{1}{2}})}^{(n)} l e t f i n d

{(x^{p})}^{(n)} w i t h p \in Q

{(x^{p})}^{(1)} = {p x}^{p - 1}, {(x^{p})}^{(2)} = p (p - 1) x^{p - 2} \Rightarrow

{(x^{p})}^{(n)} = p (p - 1) \dots (p - n + 1) x^{p - n} \Rightarrow

{(x^{- \frac{1}{2}})}^{(n)} = (- \frac{1}{2}) (- \frac{3}{2}) \dots (- \frac{1}{2} - n + 1) x^{- \frac{1}{2} - n}

= (- \frac{1}{2}) (- \frac{3}{2}) \dots (\frac{- 1 - 2 n + 2}{2}) x^{- \frac{1}{2} - n}

= (- \frac{1}{2}) (- \frac{3}{2}) \dots . (- \frac{2 n - 1}{2}) \frac{1}{x^{n} \sqrt{x}} \Rightarrow

f^{(n)} (x) = \frac{π}{\sqrt{2}} (- \frac{1}{2}) (- \frac{3}{2}) \dots (- \frac{2 n - 1}{2}) \frac{1}{x^{n} \sqrt{x}}

f^{(n)} (1) = \frac{π}{\sqrt{2}} (- \frac{1}{2}) (- \frac{3}{2}) \dots (- \frac{2 n - 1}{2})

3) f (x) = \sum_{n = 0}^{\infty} \frac{f^{(n)} (1)}{n!} {(x - 1)}^{n}

= \sum_{n = 0}^{\infty} \frac{π}{n! \sqrt{2}} (- \frac{1}{2}) (- \frac{3}{2}) \dots (- \frac{2 n - 1}{2}) {(x - 1)}^{n} .

Commented by abdo mathsup 649 cc last updated on 27/Jul/18

2) t h e Q i s f i n d f^{(n)} (x) a n d f^{(n)} (1)

3) t h e Q i s d e v e l o p p f a t i n t e g r s e r i e a t v (1) .