Дано
$$2 left(frac{sin{left (x left(frac{pi n}{5} + piright) right )}}{2 left(frac{pi n}{5} + piright)} + frac{sin{left (x left(- frac{pi n}{5} + piright) right )}}{2 left(- frac{pi n}{5} + piright)}right) + frac{25}{pi n} sin{left (frac{pi n}{5} x right )}$$
Подстановка условия
$$2 left(frac{sin{left (x left(frac{pi n}{5} + piright) right )}}{2 left(frac{pi n}{5} + piright)} + frac{sin{left (x left(- frac{pi n}{5} + piright) right )}}{2 left(- frac{pi n}{5} + piright)}right) + frac{25}{pi n} sin{left (frac{pi n}{5} x right )}$$
2*(sin((pi – n*pi/5)*(1/4))/(2*(pi – n*pi/5)) + sin(((n*pi)/5 + pi)*(1/4))/(2*(pi + (n*pi)/5))) + (25*sin(((n*pi)*(1/4))/5))/(n*pi)
$$2 left(frac{sin{left ((1/4) left(frac{pi n}{5} + piright) right )}}{2 left(frac{pi n}{5} + piright)} + frac{sin{left ((1/4) left(- frac{pi n}{5} + piright) right )}}{2 left(- frac{pi n}{5} + piright)}right) + frac{25}{pi n} sin{left (frac{pi n}{5} (1/4) right )}$$
2*(sin((pi – n*pi/5)/4)/(2*(pi – n*pi/5)) + sin(((n*pi)/5 + pi)/4)/(2*(pi + (n*pi)/5))) + (25*sin(((n*pi)/4)/5))/(n*pi)
$$2 left(frac{sin{left (frac{1}{4} left(frac{pi n}{5} + piright) right )}}{2 left(frac{pi n}{5} + piright)} + frac{sin{left (frac{1}{4} left(- frac{pi n}{5} + piright) right )}}{2 left(- frac{pi n}{5} + piright)}right) + frac{25}{pi n} sin{left (frac{pi n}{5} frac{1}{4} right )}$$
2*cos(pi/4 + pi*n/20)/(2*pi – 2*pi*n/5) + 2*sin(pi/4 + pi*n/20)/(2*pi + 2*pi*n/5) + 25*sin(pi*n/20)/(pi*n)
$$frac{2 sin{left (frac{pi n}{20} + frac{pi}{4} right )}}{frac{2 pi}{5} n + 2 pi} + frac{2 cos{left (frac{pi n}{20} + frac{pi}{4} right )}}{- frac{2 pi}{5} n + 2 pi} + frac{25}{pi n} sin{left (frac{pi n}{20} right )}$$
Степени
$$frac{2 sin{left (x left(frac{pi n}{5} + piright) right )}}{frac{2 pi}{5} n + 2 pi} + frac{2 sin{left (x left(- frac{pi n}{5} + piright) right )}}{- frac{2 pi}{5} n + 2 pi} + frac{25}{pi n} sin{left (frac{pi n}{5} x right )}$$
Численный ответ
2.0*sin(((n*pi)/5 + pi)*x)/(6.28318530717959 + 1.25663706143592*n) + 2.0*sin((pi – n*pi/5)*x)/(6.28318530717959 – 1.25663706143592*n) + 7.95774715459477*sin(((n*pi)*x)/5)/n
Рациональный знаменатель
$$frac{1}{pi n left(2 pi n – 10 piright) left(2 pi n + 10 piright)} left(100 pi^{2} n^{2} sin{left (frac{pi n}{5} x right )} + 20 pi^{2} n^{2} sin{left (frac{pi n}{5} x – pi x right )} + 20 pi^{2} n^{2} sin{left (frac{pi n}{5} x + pi x right )} + 100 pi^{2} n sin{left (frac{pi n}{5} x – pi x right )} – 100 pi^{2} n sin{left (frac{pi n}{5} x + pi x right )} – 2500 pi^{2} sin{left (frac{pi n}{5} x right )}right)$$
Объединение рациональных выражений
$$frac{1}{pi n left(- n + 5right) left(n + 5right)} left(5 n left(left(- n + 5right) sin{left (frac{pi x}{5} left(n + 5right) right )} + left(n + 5right) sin{left (frac{pi x}{5} left(- n + 5right) right )}right) + 25 left(- n + 5right) left(n + 5right) sin{left (frac{pi n}{5} x right )}right)$$
Общее упрощение
/ /pi*x*(5 + n) /pi*x*(-5 + n) /pi*n*x
5*|n*(-5 + n)*sin|————| + n*(5 + n)*sin|————-| + 5*(-5 + n)*(5 + n)*sin|——||
5 / 5 / 5 //
————————————————————————————————
pi*n*(-5 + n)*(5 + n)
$$frac{1}{pi n left(n – 5right) left(n + 5right)} left(5 n left(n – 5right) sin{left (frac{pi x}{5} left(n + 5right) right )} + 5 n left(n + 5right) sin{left (frac{pi x}{5} left(n – 5right) right )} + 25 left(n – 5right) left(n + 5right) sin{left (frac{pi n}{5} x right )}right)$$
Соберем выражение
$$frac{10 n sin{left (frac{pi n}{5} x – pi x right )}}{2 pi n^{2} – 50 pi} + frac{10 n sin{left (frac{pi n}{5} x + pi x right )}}{2 pi n^{2} – 50 pi} + frac{50 sin{left (frac{pi n}{5} x – pi x right )}}{2 pi n^{2} – 50 pi} – frac{50 sin{left (frac{pi n}{5} x + pi x right )}}{2 pi n^{2} – 50 pi} + frac{25}{pi n} sin{left (frac{pi n}{5} x right )}$$
Общий знаменатель
/pi*n*x / pi*n*x 2 / pi*n*x 2 / pi*n*x / pi*n*x 2 /pi*n*x
– 625*sin|——| – 25*n*sin|pi*x + ——| + 5*n *sin|pi*x + ——| + 5*n *sin|-pi*x + ——| + 25*n*sin|-pi*x + ——| + 25*n *sin|——|
5 / 5 / 5 / 5 / 5 / 5 /
———————————————————————————————————————————————–
3
pi*n – 25*pi*n
$$frac{1}{pi n^{3} – 25 pi n} left(25 n^{2} sin{left (frac{pi n}{5} x right )} + 5 n^{2} sin{left (frac{pi n}{5} x – pi x right )} + 5 n^{2} sin{left (frac{pi n}{5} x + pi x right )} + 25 n sin{left (frac{pi n}{5} x – pi x right )} – 25 n sin{left (frac{pi n}{5} x + pi x right )} – 625 sin{left (frac{pi n}{5} x right )}right)$$
Тригонометрическая часть
/ / n /pi*n*x
sin|pi*x*|1 + -|| 25*sin|——|
5// 1 // n*pi 5 /
—————– + 2*————-*sin||pi – —-|*x| + ————–
n*pi / n*pi 5 / / pi*n
pi + —- 2*|pi – —-|
5 5 /
$$2 frac{1}{- frac{2 pi}{5} n + 2 pi} sin{left (x left(- frac{pi n}{5} + piright) right )} + frac{sin{left (pi x left(frac{n}{5} + 1right) right )}}{frac{pi n}{5} + pi} + frac{25}{pi n} sin{left (frac{pi n}{5} x right )}$$
Комбинаторика
/ /pi*n*x 2 / pi*n*x 2 / pi*n*x / pi*n*x / pi*n*x 2 /pi*n*x
5*|- 125*sin|——| + n *sin|pi*x + ——| + n *sin|-pi*x + ——| – 5*n*sin|pi*x + ——| + 5*n*sin|-pi*x + ——| + 5*n *sin|——||
5 / 5 / 5 / 5 / 5 / 5 //
——————————————————————————————————————————————–
pi*n*(-5 + n)*(5 + n)
$$frac{1}{pi n left(n – 5right) left(n + 5right)} left(25 n^{2} sin{left (frac{pi n}{5} x right )} + 5 n^{2} sin{left (frac{pi n}{5} x – pi x right )} + 5 n^{2} sin{left (frac{pi n}{5} x + pi x right )} + 25 n sin{left (frac{pi n}{5} x – pi x right )} – 25 n sin{left (frac{pi n}{5} x + pi x right )} – 625 sin{left (frac{pi n}{5} x right )}right)$$
Раскрыть выражение
/ / pi*n / / pi*n /pi*n*x
2*sin|x*|pi – —-|| 2*sin|x*|pi + —-|| 25*sin|——|
5 // 5 // 5 /
——————– + ——————– + ————–
2*pi*n 2*pi*n pi*n
2*pi – —— 2*pi + ——
5 5
$$frac{2 sin{left (x left(frac{pi n}{5} + piright) right )}}{frac{2 pi}{5} n + 2 pi} + frac{2 sin{left (x left(- frac{pi n}{5} + piright) right )}}{- frac{2 pi}{5} n + 2 pi} + frac{25}{pi n} sin{left (frac{pi n}{5} x right )}$$
// n*pi //n*pi /n*pi*x
sin||pi – —-|*x| sin||—- + pi|*x| 25*sin|——|
5 / / 5 / / 5 /
—————— + —————— + ————–
n*pi n*pi pi*n
pi – —- pi + —-
5 5
$$frac{sin{left (x left(frac{pi n}{5} + piright) right )}}{frac{pi n}{5} + pi} + frac{sin{left (x left(- frac{pi n}{5} + piright) right )}}{- frac{pi n}{5} + pi} + frac{25}{pi n} sin{left (frac{pi n}{5} x right )}$$
