Дано
$$frac{2}{sqrt{5}} log{left (frac{- 2 sqrt{5} + frac{10 sin{left (t right )}}{cos{left (t right )} + 1}}{2 sqrt{5} + frac{10 sin{left (t right )}}{cos{left (t right )} + 1}} right )}$$
Подстановка условия
$$frac{2}{sqrt{5}} log{left (frac{- 2 sqrt{5} + frac{10 sin{left (t right )}}{cos{left (t right )} + 1}}{2 sqrt{5} + frac{10 sin{left (t right )}}{cos{left (t right )} + 1}} right )}$$
(2*log(((10*sin((1)))/(cos((1)) + 1) – 2*sqrt(5))/((10*sin((1)))/(cos((1)) + 1) + 2*sqrt(5))))/sqrt(5)
$$frac{2}{sqrt{5}} log{left (frac{- 2 sqrt{5} + frac{10 sin{left ((1) right )}}{cos{left ((1) right )} + 1}}{2 sqrt{5} + frac{10 sin{left ((1) right )}}{cos{left ((1) right )} + 1}} right )}$$
(2*log(((10*sin(1))/(cos(1) + 1) – 2*sqrt(5))/((10*sin(1))/(cos(1) + 1) + 2*sqrt(5))))/sqrt(5)
$$frac{2}{sqrt{5}} log{left (frac{- 2 sqrt{5} + frac{10 sin{left (1 right )}}{cos{left (1 right )} + 1}}{2 sqrt{5} + frac{10 sin{left (1 right )}}{cos{left (1 right )} + 1}} right )}$$
2*sqrt(5)*log((-2*sqrt(5) + 10*sin(1)/(1 + cos(1)))/(2*sqrt(5) + 10*sin(1)/(1 + cos(1))))/5
$$frac{2 sqrt{5}}{5} log{left (frac{- 2 sqrt{5} + frac{10 sin{left (1 right )}}{cos{left (1 right )} + 1}}{2 sqrt{5} + frac{10 sin{left (1 right )}}{cos{left (1 right )} + 1}} right )}$$
Степени
$$frac{2 sqrt{5}}{5} log{left (frac{- 2 sqrt{5} + frac{10 sin{left (t right )}}{cos{left (t right )} + 1}}{2 sqrt{5} + frac{10 sin{left (t right )}}{cos{left (t right )} + 1}} right )}$$
Численный ответ
0.894427190999916*log(((10*sin(t))/(cos(t) + 1) – 2*sqrt(5))/((10*sin(t))/(cos(t) + 1) + 2*sqrt(5)))
Рациональный знаменатель
$$frac{2 sqrt{5}}{5} log{left (- frac{1}{left(1 – frac{5 sin^{2}{left (t right )}}{left(cos{left (t right )} + 1right)^{2}}right) left(left(cos{left (t right )} + 1right)^{2} – frac{5 left(cos{left (t right )} + 1right)^{2} sin^{2}{left (t right )}}{left(cos{left (t right )} + 1right)^{2}}right) left(cos{left (t right )} + 1 – frac{5 sin^{2}{left (t right )} cos{left (t right )}}{left(cos{left (t right )} + 1right)^{2}} – frac{5 sin^{2}{left (t right )}}{left(cos{left (t right )} + 1right)^{2}}right)} left(- 2 sqrt{5} left(1 – frac{5 sin^{2}{left (t right )}}{left(cos{left (t right )} + 1right)^{2}}right) left(left(cos{left (t right )} + 1right)^{2} – frac{5 left(cos{left (t right )} + 1right)^{2} sin^{2}{left (t right )}}{left(cos{left (t right )} + 1right)^{2}}right) sin{left (t right )} + 5 left(1 – frac{5 sin^{2}{left (t right )}}{left(cos{left (t right )} + 1right)^{2}}right) left(cos{left (t right )} + 1 – frac{5 sin^{2}{left (t right )} cos{left (t right )}}{left(cos{left (t right )} + 1right)^{2}} – frac{5 sin^{2}{left (t right )}}{left(cos{left (t right )} + 1right)^{2}}right) sin^{2}{left (t right )} + left(left(cos{left (t right )} + 1right)^{2} – frac{5 left(cos{left (t right )} + 1right)^{2} sin^{2}{left (t right )}}{left(cos{left (t right )} + 1right)^{2}}right) left(cos{left (t right )} + 1 – frac{5 sin^{2}{left (t right )} cos{left (t right )}}{left(cos{left (t right )} + 1right)^{2}} – frac{5 sin^{2}{left (t right )}}{left(cos{left (t right )} + 1right)^{2}}right)right) right )}$$
Объединение рациональных выражений
$$frac{2 sqrt{5}}{5} log{left (frac{- sqrt{5} left(cos{left (t right )} + 1right) + 5 sin{left (t right )}}{sqrt{5} left(cos{left (t right )} + 1right) + 5 sin{left (t right )}} right )}$$
Общее упрощение
/ ___
___ |5*sin(t) – / 5 *(1 + cos(t))|
2*/ 5 *log|—————————–|
| ___ |
5*sin(t) + / 5 *(1 + cos(t))/
——————————————
5
$$frac{2 sqrt{5}}{5} log{left (frac{- sqrt{5} left(cos{left (t right )} + 1right) + 5 sin{left (t right )}}{sqrt{5} left(cos{left (t right )} + 1right) + 5 sin{left (t right )}} right )}$$
Соберем выражение
$$frac{2 sqrt{5}}{5} log{left (frac{- 2 sqrt{5} + frac{10 sin{left (t right )}}{cos{left (t right )} + 1}}{2 sqrt{5} + frac{10 sin{left (t right )}}{cos{left (t right )} + 1}} right )}$$
/ ___ ___ ___ ___ ___
___ | 75*sin(t) 10*sin(2*t) 25*sin(3*t) 5*sin(3*t) 25*sin(t) / 5 *cos(3*t) 3*/ 5 *cos(t) 8*/ 5 *cos(2*t) 5*/ 5 *cos(3*t) 7*/ 5 *cos(t) |
2*/ 5 *log|- ———————————————————————————————————— + ——————————————————————————————————— + ———————————————————————————————————— + ————————————————————————————————————- + ————————————————————————————————————- + ———————————————————————————————————— + ———————————————————————————————————— + ——————————————————————————————————— + ————————————————————————————————————- + ————————————————————————————————————-|
| ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ / ___ ___ ___ ___ / ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ / ___ ___ ___ ___ / ___ ___ ___ ___ |
-150*sin(t) – 60*sin(2*t) – 40*/ 5 + 10*sin(3*t) – 30*/ 5 *cos(t) + 14*/ 5 *cos(3*t) + 24*/ 5 *cos(2*t) -75*sin(t) – 30*sin(2*t) – 20*/ 5 + 5*sin(3*t) – 15*/ 5 *cos(t) + 7*/ 5 *cos(3*t) + 12*/ 5 *cos(2*t) -150*sin(t) – 60*sin(2*t) – 40*/ 5 + 10*sin(3*t) – 30*/ 5 *cos(t) + 14*/ 5 *cos(3*t) + 24*/ 5 *cos(2*t) 2* -75*sin(t) – 30*sin(2*t) – 20*/ 5 + 5*sin(3*t) – 15*/ 5 *cos(t) + 7*/ 5 *cos(3*t) + 12*/ 5 *cos(2*t)/ 2* -75*sin(t) – 30*sin(2*t) – 20*/ 5 + 5*sin(3*t) – 15*/ 5 *cos(t) + 7*/ 5 *cos(3*t) + 12*/ 5 *cos(2*t)/ -150*sin(t) – 60*sin(2*t) – 40*/ 5 + 10*sin(3*t) – 30*/ 5 *cos(t) + 14*/ 5 *cos(3*t) + 24*/ 5 *cos(2*t) -150*sin(t) – 60*sin(2*t) – 40*/ 5 + 10*sin(3*t) – 30*/ 5 *cos(t) + 14*/ 5 *cos(3*t) + 24*/ 5 *cos(2*t) -75*sin(t) – 30*sin(2*t) – 20*/ 5 + 5*sin(3*t) – 15*/ 5 *cos(t) + 7*/ 5 *cos(3*t) + 12*/ 5 *cos(2*t) 2* -75*sin(t) – 30*sin(2*t) – 20*/ 5 + 5*sin(3*t) – 15*/ 5 *cos(t) + 7*/ 5 *cos(3*t) + 12*/ 5 *cos(2*t)/ 2* -75*sin(t) – 30*sin(2*t) – 20*/ 5 + 5*sin(3*t) – 15*/ 5 *cos(t) + 7*/ 5 *cos(3*t) + 12*/ 5 *cos(2*t)//
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5
$$frac{2 sqrt{5}}{5} log{left (frac{25 sin{left (t right )}}{- 150 sin{left (t right )} – 60 sin{left (2 t right )} + 10 sin{left (3 t right )} – 30 sqrt{5} cos{left (t right )} + 24 sqrt{5} cos{left (2 t right )} + 14 sqrt{5} cos{left (3 t right )} – 40 sqrt{5}} + frac{10 sin{left (2 t right )}}{- 75 sin{left (t right )} – 30 sin{left (2 t right )} + 5 sin{left (3 t right )} – 15 sqrt{5} cos{left (t right )} + 12 sqrt{5} cos{left (2 t right )} + 7 sqrt{5} cos{left (3 t right )} – 20 sqrt{5}} + frac{5 sin{left (3 t right )}}{- 150 sin{left (t right )} – 60 sin{left (2 t right )} + 10 sin{left (3 t right )} – 30 sqrt{5} cos{left (t right )} + 24 sqrt{5} cos{left (2 t right )} + 14 sqrt{5} cos{left (3 t right )} – 40 sqrt{5}} + frac{7 sqrt{5} cos{left (t right )}}{- 150 sin{left (t right )} – 60 sin{left (2 t right )} + 10 sin{left (3 t right )} – 30 sqrt{5} cos{left (t right )} + 24 sqrt{5} cos{left (2 t right )} + 14 sqrt{5} cos{left (3 t right )} – 40 sqrt{5}} + frac{8 sqrt{5} cos{left (2 t right )}}{- 75 sin{left (t right )} – 30 sin{left (2 t right )} + 5 sin{left (3 t right )} – 15 sqrt{5} cos{left (t right )} + 12 sqrt{5} cos{left (2 t right )} + 7 sqrt{5} cos{left (3 t right )} – 20 sqrt{5}} + frac{5 sqrt{5} cos{left (3 t right )}}{- 150 sin{left (t right )} – 60 sin{left (2 t right )} + 10 sin{left (3 t right )} – 30 sqrt{5} cos{left (t right )} + 24 sqrt{5} cos{left (2 t right )} + 14 sqrt{5} cos{left (3 t right )} – 40 sqrt{5}} – frac{75 sin{left (t right )}}{- 150 sin{left (t right )} – 60 sin{left (2 t right )} + 10 sin{left (3 t right )} – 30 sqrt{5} cos{left (t right )} + 24 sqrt{5} cos{left (2 t right )} + 14 sqrt{5} cos{left (3 t right )} – 40 sqrt{5}} + frac{25 sin{left (3 t right )}}{- 150 sin{left (t right )} – 60 sin{left (2 t right )} + 10 sin{left (3 t right )} – 30 sqrt{5} cos{left (t right )} + 24 sqrt{5} cos{left (2 t right )} + 14 sqrt{5} cos{left (3 t right )} – 40 sqrt{5}} + frac{3 sqrt{5} cos{left (t right )}}{- 150 sin{left (t right )} – 60 sin{left (2 t right )} + 10 sin{left (3 t right )} – 30 sqrt{5} cos{left (t right )} + 24 sqrt{5} cos{left (2 t right )} + 14 sqrt{5} cos{left (3 t right )} – 40 sqrt{5}} + frac{sqrt{5} cos{left (3 t right )}}{- 150 sin{left (t right )} – 60 sin{left (2 t right )} + 10 sin{left (3 t right )} – 30 sqrt{5} cos{left (t right )} + 24 sqrt{5} cos{left (2 t right )} + 14 sqrt{5} cos{left (3 t right )} – 40 sqrt{5}} right )}$$
/ 2
|/ ___ ___ |
___ | – / 5 + 5*sin(t) – / 5 *cos(t)/ |
/ 5 *log|————————————|
| 2 |
| / ___ ___ |
/ 5 + 5*sin(t) + / 5 *cos(t)/ /
———————————————–
5
$$frac{sqrt{5}}{5} log{left (frac{left(5 sin{left (t right )} – sqrt{5} cos{left (t right )} – sqrt{5}right)^{2}}{left(5 sin{left (t right )} + sqrt{5} cos{left (t right )} + sqrt{5}right)^{2}} right )}$$
Общий знаменатель
/ ___
___ | / 5 5*sin(t) |
2*/ 5 *log|- —————— + —————————————————|
| ___ 5*sin(t) ___ ___ 5*sin(t) 5*cos(t)*sin(t)|
| / 5 + ———- / 5 + / 5 *cos(t) + ———- + —————|
1 + cos(t) 1 + cos(t) 1 + cos(t) /
—————————————————————————————
5
$$frac{2 sqrt{5}}{5} log{left (frac{5 sin{left (t right )}}{sqrt{5} cos{left (t right )} + sqrt{5} + frac{5 sin{left (t right )} cos{left (t right )}}{cos{left (t right )} + 1} + frac{5 sin{left (t right )}}{cos{left (t right )} + 1}} – frac{sqrt{5}}{sqrt{5} + frac{5 sin{left (t right )}}{cos{left (t right )} + 1}} right )}$$
Тригонометрическая часть
/ ___ 10*sin(t)
|- 2*/ 5 + ———-|
___ | cos(t) + 1|
2*/ 5 *log|———————-|
| ___ 10*sin(t) |
| 2*/ 5 + ———- |
cos(t) + 1 /
———————————–
5
$$frac{2 sqrt{5}}{5} log{left (frac{- 2 sqrt{5} + frac{10 sin{left (t right )}}{cos{left (t right )} + 1}}{2 sqrt{5} + frac{10 sin{left (t right )}}{cos{left (t right )} + 1}} right )}$$
Комбинаторика
/ ___
___ | 2*/ 5 10*sin(t) |
2*/ 5 *log|- ——————– + ——————————————————–|
| ___ 10*sin(t) ___ ___ 10*sin(t) 10*cos(t)*sin(t)|
| 2*/ 5 + ———- 2*/ 5 + 2*/ 5 *cos(t) + ———- + —————-|
1 + cos(t) 1 + cos(t) 1 + cos(t) /
———————————————————————————————-
5
$$frac{2 sqrt{5}}{5} log{left (frac{10 sin{left (t right )}}{2 sqrt{5} cos{left (t right )} + 2 sqrt{5} + frac{10 sin{left (t right )} cos{left (t right )}}{cos{left (t right )} + 1} + frac{10 sin{left (t right )}}{cos{left (t right )} + 1}} – frac{2 sqrt{5}}{2 sqrt{5} + frac{10 sin{left (t right )}}{cos{left (t right )} + 1}} right )}$$
Раскрыть выражение
___ / / ___ 10*sin(t) / ___ 10*sin(t)
/ 5 *|- 2*log|2*/ 5 + ———-| + 2*log|- 2*/ 5 + ———-||
1 + cos(t)/ 1 + cos(t)//
———————————————————————
5
$$frac{sqrt{5}}{5} left(2 log{left (- 2 sqrt{5} + frac{10 sin{left (t right )}}{cos{left (t right )} + 1} right )} – 2 log{left (2 sqrt{5} + frac{10 sin{left (t right )}}{cos{left (t right )} + 1} right )}right)$$
/ ___ 10*sin(t)
|- 2*/ 5 + ———-|
___ | 1 + cos(t)|
2*/ 5 *log|———————-|
| ___ 10*sin(t) |
| 2*/ 5 + ———- |
1 + cos(t) /
———————————–
5
$$frac{2 sqrt{5}}{5} log{left (frac{- 2 sqrt{5} + frac{10 sin{left (t right )}}{cos{left (t right )} + 1}}{2 sqrt{5} + frac{10 sin{left (t right )}}{cos{left (t right )} + 1}} right )}$$
