calculus-find-0-pi-2-dx-sin-10-x-cos-10-x-

Question Number 133688 by mnjuly1970 last updated on 23/Feb/21

You can't use 'macro parameter character #' in math mode

f i n d :

ϕ = \int_{0}^{\frac{π}{2}} \frac{d x}{{s i n}^{10} (x) + {c o s}^{10} (x)} = ? ? ?

Answered by Ar Brandon last updated on 23/Feb/21

ϕ = \int_{0}^{\frac{π}{2}} \frac{d x}{\sin^{10} x + \cos^{10} x} = \int_{0}^{\frac{π}{2}} \frac{\sec^{10} x}{\tan^{10} x + 1} d x = \int_{0}^{\infty} \frac{{(t^{2} + 1)}^{4}}{t^{10} + 1} dt

= \int_{0}^{\infty} {\frac{t^{8} + {4 t}^{6} + {6 t}^{4} + {4 t}^{2} + 1}{t^{10} + 1}} dt, u = t^{10} \Rightarrow du = {10 t}^{9} dt = {10 u}^{\frac{9}{10}} dt

ϕ = \frac{1}{10} \int_{0}^{\infty} \frac{u^{- \frac{1}{10}}}{u + 1} du + \frac{2}{5} \int_{0}^{\infty} \frac{u^{- \frac{3}{10}}}{u + 1} du + \frac{3}{5} \int_{0}^{\infty} \frac{u^{- \frac{5}{10}}}{u + 1} du + \frac{2}{5} \int_{0}^{\infty} \frac{u^{- \frac{7}{10}}}{u + 1} du + \frac{1}{10} \int_{0}^{\infty} \frac{u^{- \frac{9}{10}}}{u + 1} du

= \frac{1}{10} β (\frac{9}{10}, \frac{1}{10}) + \frac{2}{5} β (\frac{7}{10}, \frac{3}{10}) + \frac{3}{5} β (\frac{1}{2}, \frac{1}{2}) + \frac{2}{5} β (\frac{3}{10}, \frac{7}{10}) + \frac{1}{10} β (\frac{1}{10}, \frac{9}{10})

= \frac{π}{5 \sin (\frac{π}{10})} + \frac{4 π}{5 \sin (\frac{3 π}{10})} + \frac{3 π}{5 \sin (\frac{π}{2})}

Commented by Dwaipayan Shikari last updated on 23/Feb/21

\frac{π}{5} (\frac{4}{\sqrt{5} - 1} + \frac{16}{\sqrt{5} + 1} + 3) = \frac{π}{5} (\sqrt{5} + 1 + 4 \sqrt{5} - 4 + 3)

= \sqrt{5} π

Commented by mnjuly1970 last updated on 23/Feb/21

t h a n k s a l o t m a s t e r b r a n d o n

n i c e v e r y n i c e s o l u t i o n \dots

Commented by Ar Brandon last updated on 23/Feb/21

My pleasure Sir

Answered by Ajetunmobi last updated on 23/Feb/21

i have did this on facebook

Answered by Ajetunmobi last updated on 23/Feb/21

Commented by Ar Brandon last updated on 23/Feb/21

Ambiguity on your name Sir. Ajetunmobi or Ajeutunmobi ?��

Commented by Ajetunmobi last updated on 24/Feb/21

i t i s Ajetunmobi Abdulqoyyum

thank u that was a mistake

Answered by Ajetunmobi last updated on 23/Feb/21