0-pi-sin-21x-2-sin-x-2-dx-

Question Number 85677 by john santu last updated on 24/Mar/20

\underset{0}{\int^{π}} \frac{\sin (\frac{21 x}{2})}{\sin (\frac{x}{2})} d x

Commented by jagoll last updated on 24/Mar/20

i {don}^{'} t have a simple way to solve

it

Commented by john santu last updated on 24/Mar/20

u s i n g R e d u c t i o n f o r m u l a

Commented by M±th+et£s last updated on 25/Mar/20

\int_{0}^{π} \frac{s i n (\frac{21 x}{2})}{s i n (\frac{x}{2})} d x

l e t y = \frac{x}{2}

\int_{0}^{\frac{π}{2}} \frac{(s i n 21 y)}{s i n (y)} 2 d y

2 \int_{0}^{\frac{π}{2}} \frac{s i n 20 y c o s (y) + c o s 20 y s i n y}{s i n y} d y

2 \int_{0}^{\frac{π}{2}} s i n 20 y c o t y d y + \int_{0}^{\frac{π}{2}} c o s 20 y d y

\int_{0}^{\frac{π}{2}} c o s y d y = 0

2 \int_{0}^{\frac{π}{2}} s i n 20 y c o t y d y = 2 \int_{0}^{\frac{π}{2}} (1 + 2 c o s 2 y + 2 c o s 4 y + \dots + c o s 20 y) d y

{[y + s i n 2 y + \frac{1}{2} s i n 4 y + \frac{1}{3} s i n 6 y + \dots]}_{0}^{\frac{π}{2}}

π + 0 + \dots .

s o \int_{0}^{π} \frac{s i n (\frac{21 x}{2})}{s i n (\frac{x}{2})} d x = π

Answered by mind is power last updated on 24/Mar/20

\underset{k = 0}{\sum^{20}} e^{i k x} = \frac{e^{i 10 x} s i n (\frac{21 x}{2})}{s i n (\frac{x}{2})}

\Rightarrow \int_{0}^{π} \frac{s i n (\frac{21 x}{2})}{s i n (\frac{x}{2})} d x = \underset{k . 0}{\sum^{20}} \int_{0}^{π} e^{i (k - 10) x} d x e a s y n o w

= π + \underset{k = 0}{\sum^{9}} \int_{0}^{π} (e^{i (k - 10) x}) d x + \underset{k = 0}{\sum^{9}} \int_{0}^{π} (e^{i (1 + k) x}) d x

= π + \underset{k = 0}{\sum^{9}} {\frac{1}{i (k - 10)} [e^{i (k - 10) π} - 1] + \frac{e^{i (1 + k) π} - 1}{i (1 + k)}}

= π + \underset{k = 0}{\sum^{9}} \frac{{(- 1)}^{k} - 1}{i (k - 10)} + \underset{k = 0}{\sum^{9}} \frac{{(- 1)}^{k} - 1}{i (1 + k)}

= π - \underset{k = 0}{\sum^{9}} \frac{{(- 1)}^{k} - 1}{i (1 + k)} + \underset{k = 0}{\sum^{9}} \frac{{(- 1)}^{k} - 1}{i (1 + k)} = π

Commented by jagoll last updated on 25/Mar/20

sir what it formula ?

Commented by mind is power last updated on 26/Mar/20

F o r m u l a ?

Answered by john santu last updated on 25/Mar/20

I = \underset{0}{\int^{π}} \frac{\sin (\frac{21 x}{2})}{\sin (\frac{x}{2})} d x, [t = \frac{x}{2}]

I = 2 \underset{0}{\int^{\frac{π}{2}}} \frac{\sin 21 t}{\sin t} d t =

c o n s i d e r G_{n} = \underset{0}{\int^{\frac{π}{2}}} \frac{\sin n t}{\sin t} d t

t h e n G_{n + 2} - G_{n} = \underset{0}{\int^{\frac{π}{2}}} \frac{\sin ((n + 2) t) - \sin (n t)}{\sin t} d t

= \int \underset{0}{\overset{\frac{π}{2}}{}} \frac{\sin t \cos ((n + 1) t)}{\sin t} d t

= \frac{\sin (\frac{(n + 1) π}{2})}{n + 1}

G_{n + 2} = G_{n} + \frac{\sin (\frac{(n + 1) π}{2})}{n + 1}, n ⩾ 0

w h e n n i s o d d n = 2 k - 1

G_{2 k + 1} = G_{2 k - 1}, k ⩾ 1

G_{21} = G_{1} = {\int_{0}}^{\frac{π}{2}} \frac{\sin t}{\sin t} d t = \frac{π}{2}

s o t h a t

I = \underset{0}{\int^{π}} \frac{\sin (\frac{21 x}{2})}{\sin (\frac{x}{2})} d x = 2 G_{21} = 2 \times \frac{π}{2}

= π