5*(-sin(x)-720*cos(x)*1/(x^5)-360*sin(x)*1/(x^4)-6*cos(x)*1/x+30*sin(x)*1/(x^2)+120*cos(x)*1/(x^3)+720*sin(x)*1/(x^6))*1/x если x=1 (упростите выражение)

Дано

$$\frac{5}{x} \left(- \sin{\left (x \right )} — \frac{720}{x^{5}} \cos{\left (x \right )} — \frac{360}{x^{4}} \sin{\left (x \right )} — \frac{6}{x} \cos{\left (x \right )} + \frac{30}{x^{2}} \sin{\left (x \right )} + \frac{120}{x^{3}} \cos{\left (x \right )} + \frac{720}{x^{6}} \sin{\left (x \right )}\right)$$
Подстановка условия
$$\frac{5}{x} \left(- \sin{\left (x \right )} — \frac{720}{x^{5}} \cos{\left (x \right )} — \frac{360}{x^{4}} \sin{\left (x \right )} — \frac{6}{x} \cos{\left (x \right )} + \frac{30}{x^{2}} \sin{\left (x \right )} + \frac{120}{x^{3}} \cos{\left (x \right )} + \frac{720}{x^{6}} \sin{\left (x \right )}\right)$$

(5*(-sin((1)) — 720*cos((1))/(1)^5 — 360*sin((1))/(1)^4 — 6*cos((1))/(1) + (30*sin((1)))/(1)^2 + (120*cos((1)))/(1)^3 + (720*sin((1)))/(1)^6))/(1)

$$\frac{5}{(1)} \left(- \sin{\left ((1) \right )} — \frac{720}{(1)^{5}} \cos{\left ((1) \right )} — \frac{360}{(1)^{4}} \sin{\left ((1) \right )} — \frac{6}{(1)} \cos{\left ((1) \right )} + \frac{30}{(1)^{2}} \sin{\left ((1) \right )} + \frac{120}{(1)^{3}} \cos{\left ((1) \right )} + \frac{720}{(1)^{6}} \sin{\left ((1) \right )}\right)$$

(5*(-sin(1) — 720*cos(1)/1^5 — 360*sin(1)/1^4 — 6*cos(1)/1 + (30*sin(1))/1^2 + (120*cos(1))/1^3 + (720*sin(1))/1^6))/1

/ 720*cos(1) 360*sin(1) 6*cos(1) 30*sin(1) 120*cos(1) 720*sin(1)
5*|-sin(1) — ———- — ———- — ——— + ——— + ———- + ———-|
| 5 4 1 2 3 6 |
1 1 1 1 1 /
—————————————————————————————
1

-3030*cos(1) + 1945*sin(1)

$$- 3030 \cos{\left (1 \right )} + 1945 \sin{\left (1 \right )}$$
Степени
$$\frac{1}{x} \left(- 5 \sin{\left (x \right )} — \frac{30}{x} \cos{\left (x \right )} + \frac{150}{x^{2}} \sin{\left (x \right )} + \frac{600}{x^{3}} \cos{\left (x \right )} — \frac{1800}{x^{4}} \sin{\left (x \right )} — \frac{3600}{x^{5}} \cos{\left (x \right )} + \frac{3600}{x^{6}} \sin{\left (x \right )}\right)$$
Читайте также  5^(x-1)+5*(1/5)^(x-2)=26
Численный ответ

5.0*(-sin(x) + 30.0*sin(x)/x^2 + 120.0*cos(x)/x^3 + 720.0*sin(x)/x^6 — 6.0*cos(x)/x — 360.0*sin(x)/x^4 — 720.0*cos(x)/x^5)/x

Рациональный знаменатель
$$\frac{1}{x^{22}} \left(3600 x^{15} \sin{\left (x \right )} + 5 x^{6} \left(120 x^{12} \cos{\left (x \right )} + x^{3} \left(30 x^{10} \sin{\left (x \right )} + x^{2} \left(- 6 x^{9} \cos{\left (x \right )} + x \left(- 360 x^{5} \sin{\left (x \right )} + x^{4} \left(- x^{5} \sin{\left (x \right )} — 720 \cos{\left (x \right )}\right)\right)\right)\right)\right)\right)$$
Объединение рациональных выражений
$$\frac{1}{x^{7}} \left(5 x \left(- x^{5} \sin{\left (x \right )} — 6 x^{4} \cos{\left (x \right )} + 30 x^{3} \sin{\left (x \right )} + 120 x^{2} \cos{\left (x \right )} — 360 x \sin{\left (x \right )} — 720 \cos{\left (x \right )}\right) + 3600 \sin{\left (x \right )}\right)$$
Общее упрощение

/ 6 2 5 4 3
5*720*sin(x) — x *sin(x) — 720*x*cos(x) — 360*x *sin(x) — 6*x *cos(x) + 30*x *sin(x) + 120*x *cos(x)/
——————————————————————————————————
7
x

$$\frac{1}{x^{7}} \left(- 5 x^{6} \sin{\left (x \right )} — 30 x^{5} \cos{\left (x \right )} + 150 x^{4} \sin{\left (x \right )} + 600 x^{3} \cos{\left (x \right )} — 1800 x^{2} \sin{\left (x \right )} — 3600 x \cos{\left (x \right )} + 3600 \sin{\left (x \right )}\right)$$
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Соберем выражение
$$\frac{1}{x} \left(- 5 \sin{\left (x \right )} + \frac{3600}{x^{6}} \sin{\left (x \right )} — \frac{3600}{x^{5}} \cos{\left (x \right )} — \frac{1800}{x^{4}} \sin{\left (x \right )} + \frac{600}{x^{3}} \cos{\left (x \right )} + \frac{150}{x^{2}} \sin{\left (x \right )} — \frac{30}{x} \cos{\left (x \right )}\right)$$

3600*cos(x) 1800*sin(x) 30*cos(x) 5*sin(x) 150*sin(x) 600*cos(x) 3600*sin(x)
— ———— — ———— — ——— — ——— + ———- + ———- + ————
6 5 2 x 3 4 7
x x x x x x

$$- \frac{5}{x} \sin{\left (x \right )} — \frac{30}{x^{2}} \cos{\left (x \right )} + \frac{150}{x^{3}} \sin{\left (x \right )} + \frac{600}{x^{4}} \cos{\left (x \right )} — \frac{1800}{x^{5}} \sin{\left (x \right )} — \frac{3600}{x^{6}} \cos{\left (x \right )} + \frac{3600}{x^{7}} \sin{\left (x \right )}$$
Общий знаменатель

/ 3 4 6 5 2
— -3600*sin(x) — 600*x *cos(x) — 150*x *sin(x) + 5*x *sin(x) + 30*x *cos(x) + 1800*x *sin(x) + 3600*x*cos(x)/
—————————————————————————————————————
7
x

$$- \frac{1}{x^{7}} \left(5 x^{6} \sin{\left (x \right )} + 30 x^{5} \cos{\left (x \right )} — 150 x^{4} \sin{\left (x \right )} — 600 x^{3} \cos{\left (x \right )} + 1800 x^{2} \sin{\left (x \right )} + 3600 x \cos{\left (x \right )} — 3600 \sin{\left (x \right )}\right)$$
Комбинаторика

/ 6 3 4 5 2
-5* -720*sin(x) + x *sin(x) — 120*x *cos(x) — 30*x *sin(x) + 6*x *cos(x) + 360*x *sin(x) + 720*x*cos(x)/
———————————————————————————————————
7
x

$$- \frac{1}{x^{7}} \left(5 x^{6} \sin{\left (x \right )} + 30 x^{5} \cos{\left (x \right )} — 150 x^{4} \sin{\left (x \right )} — 600 x^{3} \cos{\left (x \right )} + 1800 x^{2} \sin{\left (x \right )} + 3600 x \cos{\left (x \right )} — 3600 \sin{\left (x \right )}\right)$$
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Тригонометрическая часть

3600*cos(x) 1800*sin(x) 30*cos(x) 150*sin(x) 600*cos(x) 3600*sin(x)
-5*sin(x) — ———— — ———— — ——— + ———- + ———- + ————
5 4 x 2 3 6
x x x x x
——————————————————————————————
x

$$\frac{1}{x} \left(- 5 \sin{\left (x \right )} — \frac{30}{x} \cos{\left (x \right )} + \frac{150}{x^{2}} \sin{\left (x \right )} + \frac{600}{x^{3}} \cos{\left (x \right )} — \frac{1800}{x^{4}} \sin{\left (x \right )} — \frac{3600}{x^{5}} \cos{\left (x \right )} + \frac{3600}{x^{6}} \sin{\left (x \right )}\right)$$
Раскрыть выражение

3600*cos(x) 1800*sin(x) 30*cos(x) 150*sin(x) 600*cos(x) 3600*sin(x)
-5*sin(x) — ———— — ———— — ——— + ———- + ———- + ————
5 4 x 2 3 6
x x x x x
——————————————————————————————
x

$$\frac{1}{x} \left(- 5 \sin{\left (x \right )} — \frac{30}{x} \cos{\left (x \right )} + \frac{150}{x^{2}} \sin{\left (x \right )} + \frac{600}{x^{3}} \cos{\left (x \right )} — \frac{1800}{x^{4}} \sin{\left (x \right )} — \frac{3600}{x^{5}} \cos{\left (x \right )} + \frac{3600}{x^{6}} \sin{\left (x \right )}\right)$$
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