Дано
$$frac{1}{50} log{left (left(- 8 sin{left (50 x right )} – 2right) cos{left (100 x right )} + sin^{2}{left (100 x right )} + sin{left (100 x right )} 8 cos{left (50 x right )} + cos^{2}{left (100 x right )} + 16 sin^{2}{left (50 x right )} + 8 sin{left (50 x right )} + 16 cos^{2}{left (50 x right )} + 1 right )}$$
Подстановка условия
$$frac{1}{50} log{left (left(- 8 sin{left (50 x right )} – 2right) cos{left (100 x right )} + sin^{2}{left (100 x right )} + sin{left (100 x right )} 8 cos{left (50 x right )} + cos^{2}{left (100 x right )} + 16 sin^{2}{left (50 x right )} + 8 sin{left (50 x right )} + 16 cos^{2}{left (50 x right )} + 1 right )}$$
log(sin(100*(-4))^2 + (8*cos(50*(-4)))*sin(100*(-4)) + cos(100*(-4))^2 + (-2 – 8*sin(50*(-4)))*cos(100*(-4)) + 16*sin(50*(-4))^2 + 8*sin(50*(-4)) + 16*cos(50*(-4))^2 + 1)/50
$$frac{1}{50} log{left (left(- 8 sin{left (50 (-4) right )} – 2right) cos{left (100 (-4) right )} + sin^{2}{left (100 (-4) right )} + sin{left (100 (-4) right )} 8 cos{left (50 (-4) right )} + cos^{2}{left (100 (-4) right )} + 16 sin^{2}{left (50 (-4) right )} + 8 sin{left (50 (-4) right )} + 16 cos^{2}{left (50 (-4) right )} + 1 right )}$$
log(sin(100*(-4))^2 + (8*cos(50*(-4)))*sin(100*(-4)) + cos(100*(-4))^2 + (-2 – 8*sin(50*(-4)))*cos(100*(-4)) + 16*sin(50*(-4))^2 + 8*sin(50*(-4)) + 16*cos(50*(-4))^2 + 1)/50
$$frac{1}{50} log{left (1 + 16 cos^{2}{left (-4 cdot 50 right )} + 8 sin{left (-4 cdot 50 right )} + cos^{2}{left (-4 cdot 100 right )} + sin^{2}{left (-4 cdot 100 right )} + sin{left (-4 cdot 100 right )} 8 cos{left (-4 cdot 50 right )} + left(- 8 sin{left (-4 cdot 50 right )} – 2right) cos{left (-4 cdot 100 right )} + 16 sin^{2}{left (-4 cdot 50 right )} right )}$$
log(1 + cos(400)^2 + sin(400)^2 – 8*sin(200) + 16*cos(200)^2 + 16*sin(200)^2 + (-2 + 8*sin(200))*cos(400) – 8*cos(200)*sin(400))/50
$$frac{1}{50} log{left (cos^{2}{left (400 right )} + sin^{2}{left (400 right )} + 1 – 8 sin{left (400 right )} cos{left (200 right )} + 16 cos^{2}{left (200 right )} + left(8 sin{left (200 right )} – 2right) cos{left (400 right )} – 8 sin{left (200 right )} + 16 sin^{2}{left (200 right )} right )}$$
Степени
$$frac{1}{50} log{left (left(- 8 sin{left (50 x right )} – 2right) cos{left (100 x right )} + 16 sin^{2}{left (50 x right )} + 8 sin{left (50 x right )} + sin^{2}{left (100 x right )} + 8 sin{left (100 x right )} cos{left (50 x right )} + 16 cos^{2}{left (50 x right )} + cos^{2}{left (100 x right )} + 1 right )}$$
Численный ответ
0.02*log(sin(100*x)^2 + (8*cos(50*x))*sin(100*x) + cos(100*x)^2 + (-2 – 8*sin(50*x))*cos(100*x) + 16*sin(50*x)^2 + 8*sin(50*x) + 16*cos(50*x)^2 + 1)
Рациональный знаменатель
$$frac{1}{50} log{left (16 sin^{2}{left (50 x right )} – 8 sin{left (50 x right )} cos{left (100 x right )} + 8 sin{left (50 x right )} + sin^{2}{left (100 x right )} + 8 sin{left (100 x right )} cos{left (50 x right )} + 16 cos^{2}{left (50 x right )} + cos^{2}{left (100 x right )} – 2 cos{left (100 x right )} + 1 right )}$$
Объединение рациональных выражений
$$frac{1}{50} log{left (2 left(- 4 sin{left (50 x right )} – 1right) cos{left (100 x right )} + left(sin{left (100 x right )} + 8 cos{left (50 x right )}right) sin{left (100 x right )} + 16 sin^{2}{left (50 x right )} + 8 sin{left (50 x right )} + 16 cos^{2}{left (50 x right )} + cos^{2}{left (100 x right )} + 1 right )}$$
Общее упрощение
/ 2
log24 – 8*cos(50*x) – 4*(1 – cos(50*x)) + 16*sin(50*x)/
———————————————————
50
$$frac{1}{50} log{left (- 4 left(- cos{left (50 x right )} + 1right)^{2} + 16 sin{left (50 x right )} – 8 cos{left (50 x right )} + 24 right )}$$
Соберем выражение
$$frac{1}{50} log{left (16 sin{left (50 x right )} – 2 cos{left (100 x right )} + 18 right )}$$
/ 2 2 2 2
log1 + cos (100*x) + sin (100*x) + 8*sin(50*x) + 16*cos (50*x) + 16*sin (50*x) + (-2 – 8*sin(50*x))*cos(100*x) + 8*cos(50*x)*sin(100*x)/
—————————————————————————————————————————————–
50
$$frac{1}{50} log{left (left(- 8 sin{left (50 x right )} – 2right) cos{left (100 x right )} + 16 sin^{2}{left (50 x right )} + 8 sin{left (50 x right )} + sin^{2}{left (100 x right )} + 8 sin{left (100 x right )} cos{left (50 x right )} + 16 cos^{2}{left (50 x right )} + cos^{2}{left (100 x right )} + 1 right )}$$
Общий знаменатель
/ 2 2 2 2
log1 + cos (100*x) + sin (100*x) – 2*cos(100*x) + 8*sin(50*x) + 16*cos (50*x) + 16*sin (50*x) – 8*cos(100*x)*sin(50*x) + 8*cos(50*x)*sin(100*x)/
————————————————————————————————————————————————-
50
$$frac{1}{50} log{left (16 sin^{2}{left (50 x right )} – 8 sin{left (50 x right )} cos{left (100 x right )} + 8 sin{left (50 x right )} + sin^{2}{left (100 x right )} + 8 sin{left (100 x right )} cos{left (50 x right )} + 16 cos^{2}{left (50 x right )} + cos^{2}{left (100 x right )} – 2 cos{left (100 x right )} + 1 right )}$$
Тригонометрическая часть
/ 2
log24 – 8*cos(50*x) – 4*(1 – cos(50*x)) + 16*sin(50*x)/
———————————————————
50
$$frac{1}{50} log{left (- 4 left(- cos{left (50 x right )} + 1right)^{2} + 16 sin{left (50 x right )} – 8 cos{left (50 x right )} + 24 right )}$$
Комбинаторика
/ 2 2 2 2
log1 + cos (100*x) + sin (100*x) – 2*cos(100*x) + 8*sin(50*x) + 16*cos (50*x) + 16*sin (50*x) – 8*cos(100*x)*sin(50*x) + 8*cos(50*x)*sin(100*x)/
————————————————————————————————————————————————-
50
$$frac{1}{50} log{left (16 sin^{2}{left (50 x right )} – 8 sin{left (50 x right )} cos{left (100 x right )} + 8 sin{left (50 x right )} + sin^{2}{left (100 x right )} + 8 sin{left (100 x right )} cos{left (50 x right )} + 16 cos^{2}{left (50 x right )} + cos^{2}{left (100 x right )} – 2 cos{left (100 x right )} + 1 right )}$$
