Дано
$$2 left(20 left(5 x – 3right) cos{left (x right )} sin{left (x right )} + – left(5 x – 3right)^{2} sin^{2}{left (x right )} + left(5 x – 3right)^{2} cos^{2}{left (x right )} + 25 sin^{2}{left (x right )}right)$$
Подстановка условия
$$2 left(20 left(5 x – 3right) cos{left (x right )} sin{left (x right )} + – left(5 x – 3right)^{2} sin^{2}{left (x right )} + left(5 x – 3right)^{2} cos^{2}{left (x right )} + 25 sin^{2}{left (x right )}right)$$
2*(25*sin((1))^2 + (-3 + 5*(1))^2*cos((1))^2 – (-3 + 5*(1))^2*sin((1))^2 + ((20*(-3 + 5*(1)))*cos((1)))*sin((1)))
$$2 left(20 left(5 (1) – 3right) cos{left ((1) right )} sin{left ((1) right )} + – left(5 (1) – 3right)^{2} sin^{2}{left ((1) right )} + left(5 (1) – 3right)^{2} cos^{2}{left ((1) right )} + 25 sin^{2}{left ((1) right )}right)$$
2*(25*sin(1)^2 + (-3 + 5)^2*cos(1)^2 – (-3 + 5)^2*sin(1)^2 + ((20*(-3 + 5))*cos(1))*sin(1))
$$2 left(- left(-3 + 5right)^{2} sin^{2}{left (1 right )} + left(-3 + 5right)^{2} cos^{2}{left (1 right )} + 25 sin^{2}{left (1 right )} + 20 left(-3 + 5right) cos{left (1 right )} sin{left (1 right )}right)$$
8*cos(1)^2 + 42*sin(1)^2 + 80*cos(1)*sin(1)
$$8 cos^{2}{left (1 right )} + 42 sin^{2}{left (1 right )} + 80 sin{left (1 right )} cos{left (1 right )}$$
Степени
$$- 2 left(5 x – 3right)^{2} sin^{2}{left (x right )} + 2 left(5 x – 3right)^{2} cos^{2}{left (x right )} + 2 left(100 x – 60right) sin{left (x right )} cos{left (x right )} + 50 sin^{2}{left (x right )}$$
Численный ответ
50.0*sin(x)^2 + 2.0*(-3.0 + 5.0*x)^2*cos(x)^2 – 2.0*(-3.0 + 5.0*x)^2*sin(x)^2 + 40.0*(-3.0 + 5.0*x)*cos(x)*sin(x)
Рациональный знаменатель
$$- 2 left(5 x – 3right)^{2} sin^{2}{left (x right )} + 2 left(5 x – 3right)^{2} cos^{2}{left (x right )} + 2 left(100 x – 60right) sin{left (x right )} cos{left (x right )} + 50 sin^{2}{left (x right )}$$
Объединение рациональных выражений
$$- 2 left(5 x – 3right)^{2} sin^{2}{left (x right )} + 2 left(5 x – 3right)^{2} cos^{2}{left (x right )} + 40 left(5 x – 3right) sin{left (x right )} cos{left (x right )} + 50 sin^{2}{left (x right )}$$
Общее упрощение
2 2 2 2 2
50*sin (x) + (-60 + 100*x)*sin(2*x) – 2*(-3 + 5*x) *sin (x) + 2*(-3 + 5*x) *cos (x)
$$- 2 left(5 x – 3right)^{2} sin^{2}{left (x right )} + 2 left(5 x – 3right)^{2} cos^{2}{left (x right )} + left(100 x – 60right) sin{left (2 x right )} + 50 sin^{2}{left (x right )}$$
Соберем выражение
$$2 cdot 20 left(5 x – 3right) cos{left (x right )} sin{left (x right )} – 2 left(5 x – 3right)^{2} sin^{2}{left (x right )} + 2 left(5 x – 3right)^{2} cos^{2}{left (x right )} + 50 sin^{2}{left (x right )}$$
2
25 – 60*sin(2*x) – 7*cos(2*x) – 60*x*cos(2*x) + 50*x *cos(2*x) + 100*x*sin(2*x)
$$50 x^{2} cos{left (2 x right )} + 100 x sin{left (2 x right )} – 60 x cos{left (2 x right )} – 60 sin{left (2 x right )} – 7 cos{left (2 x right )} + 25$$
Комбинаторика
/ 2 2 2 2 2 2 2 2
2*9*cos (x) + 16*sin (x) – 60*cos(x)*sin(x) – 30*x*cos (x) – 25*x *sin (x) + 25*x *cos (x) + 30*x*sin (x) + 100*x*cos(x)*sin(x)/
$$2 left(- 25 x^{2} sin^{2}{left (x right )} + 25 x^{2} cos^{2}{left (x right )} + 30 x sin^{2}{left (x right )} + 100 x sin{left (x right )} cos{left (x right )} – 30 x cos^{2}{left (x right )} + 16 sin^{2}{left (x right )} – 60 sin{left (x right )} cos{left (x right )} + 9 cos^{2}{left (x right )}right)$$
Общий знаменатель
2 2 2 2 2 2 2 2
18*cos (x) + 32*sin (x) – 120*cos(x)*sin(x) – 60*x*cos (x) – 50*x *sin (x) + 50*x *cos (x) + 60*x*sin (x) + 200*x*cos(x)*sin(x)
$$- 50 x^{2} sin^{2}{left (x right )} + 50 x^{2} cos^{2}{left (x right )} + 60 x sin^{2}{left (x right )} + 200 x sin{left (x right )} cos{left (x right )} – 60 x cos^{2}{left (x right )} + 32 sin^{2}{left (x right )} – 120 sin{left (x right )} cos{left (x right )} + 18 cos^{2}{left (x right )}$$
Тригонометрическая часть
2 2 2 2 2
50*sin (x) + (-60 + 100*x)*sin(2*x) – 2*(-3 + 5*x) *sin (x) + 2*(-3 + 5*x) *cos (x)
$$- 2 left(5 x – 3right)^{2} sin^{2}{left (x right )} + 2 left(5 x – 3right)^{2} cos^{2}{left (x right )} + left(100 x – 60right) sin{left (2 x right )} + 50 sin^{2}{left (x right )}$$
Раскрыть выражение
2 2 2 2 2
50*sin (x) – 2*(-3 + 5*x) *sin (x) + 2*(-3 + 5*x) *cos (x) + 2*(-60 + 100*x)*cos(x)*sin(x)
$$- 2 left(5 x – 3right)^{2} sin^{2}{left (x right )} + 2 left(5 x – 3right)^{2} cos^{2}{left (x right )} + 2 left(100 x – 60right) sin{left (x right )} cos{left (x right )} + 50 sin^{2}{left (x right )}$$
