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Question Number 140628 by mohammad17 last updated on 10/May/21

w i t h o u t s p e c i a l f u n c t i o n f i n d

\int_{0}^{\frac{π}{10}} \sqrt{t a n x} d x ?

Answered by Dwaipayan Shikari last updated on 10/May/21

\int \sqrt{t a n x} d x t a n x = t^{2}

= \int \frac{2 t^{2}}{1 + t^{4}} d t = 2 \int \frac{1}{(t^{2} + \frac{1}{t^{2}})} d t = \int \frac{1 - \frac{1}{t^{2}}}{{(t + \frac{1}{t})}^{2} - 2} + \frac{1 + \frac{1}{t^{2}}}{{(t - \frac{1}{t})}^{2} + 2} d t

= \frac{1}{2 \sqrt{2}} l o g (\frac{t + \frac{1}{t} - \sqrt{2}}{t + \frac{1}{t} + \sqrt{2}}) + \frac{1}{\sqrt{2}} {t a n}^{- 1} (\frac{t - \frac{1}{t}}{\sqrt{2}}) + C

= \frac{1}{2 \sqrt{2}} l o g (\frac{t^{2} - \sqrt{2} t + 1}{t^{2} + \sqrt{2} t + 1}) + \frac{1}{\sqrt{2}} {t a n}^{- 1} (\frac{t^{2} - 1}{\sqrt{2 t}}) + C

\int_{0}^{\frac{π}{10}} \sqrt{t a n x} d x = \frac{1}{2 \sqrt{2}} l o g (\frac{t a n \frac{π}{10} - \sqrt{2 t a n \frac{π}{10}} + 1}{t a n \frac{π}{10} + \sqrt{2 t a n \frac{π}{10}} + 1}) + \frac{1}{\sqrt{2}} {t a n}^{- 1} (\frac{t a n \frac{π}{10} - 1}{\sqrt{2 t a n \frac{π}{10}}}) + \frac{π}{2 \sqrt{2}}

t a n \frac{π}{10} = \frac{\sqrt{5} - 1}{\sqrt{10 + 2 \sqrt{5}}}

= \frac{1}{\sqrt{2}} l o g (\frac{\sqrt{5} - 1}{10 + 2 \sqrt{5}} - \sqrt{\frac{\sqrt{5} - 1}{5 + \sqrt{5}}} + 1) - \frac{1}{2 \sqrt{2}} l o g (\frac{8}{5 + \sqrt{5}}) + \frac{1}{\sqrt{2}} {t a n}^{- 1} (\frac{\frac{\sqrt{5} - 1}{\sqrt{10 + 2 \sqrt{5}}} - 1}{\sqrt{\frac{\sqrt{5} - 1}{5 + \sqrt{5}}}}) + \frac{π}{2 \sqrt{2}}

Answered by Dwaipayan Shikari last updated on 10/May/21

\int_{0}^{a} {s i n}^{α} (x) {c o s}^{β} (x) d x

= \int_{0}^{s i n (a)} \frac{t^{α}}{\sqrt{1 - t^{2}}} {(\sqrt{1 - t^{2}})}^{β} d t

= \int_{0}^{s i n (a)} \frac{t^{α}}{{(1 - t^{2})}^{\frac{1 - β}{2}}} d t = \sum_{n ⩾ 0} \int_{0}^{s i n (a)} \frac{{(\frac{1 - β}{2})}_{n}}{n!} t^{α + 2 n} d t

= {s i n}^{α + 1} (a) \sum_{n ⩾ 0} \frac{{(\frac{1 - β}{2})}_{n}}{n! (α + 2 n + 1)} {({s i n}^{2} (a))}^{2 n}

= \frac{1}{α + 1} {s i n}^{α + 1} (a) \sum_{n ⩾ 0} \frac{{(\frac{1 - β}{2})}_{n} {(\frac{α + 1}{2})}_{n}}{n! {(\frac{α + 3}{2})}_{n}} {({s i n}^{2} (a))}^{2 n}

= \frac{{s i n}^{α + 1} (a)}{α + 1}_{2} F_{1} (\frac{1 - β}{2}, \frac{α + 1}{2}; \frac{α + 3}{2}; {s i n}^{2} (a))

\int_{0}^{\frac{π}{10}} \sqrt{t a n x} d x = \frac{{s i n}^{3 / 2} (\frac{π}{10})}{\frac{1}{2} + 1}_{2} F_{1} (\frac{3}{4}, \frac{3}{4}; \frac{7}{4}; \frac{3 - \sqrt{5}}{8})

= \frac{2}{3} {(\frac{3 - \sqrt{5}}{8})}^{3 / 2}_{2} F_{1} (\frac{3}{4}; \frac{3}{4}; \frac{7}{4}; \frac{3 - \sqrt{5}}{8})