Integral de $$$- 8 \cos{\left(t \right)} - 1$$$
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Tu entrada
Halla $$$\int \left(- 8 \cos{\left(t \right)} - 1\right)\, dt$$$.
Solución
Integra término a término:
$${\color{red}{\int{\left(- 8 \cos{\left(t \right)} - 1\right)d t}}} = {\color{red}{\left(- \int{1 d t} - \int{8 \cos{\left(t \right)} d t}\right)}}$$
Aplica la regla de la constante $$$\int c\, dt = c t$$$ con $$$c=1$$$:
$$- \int{8 \cos{\left(t \right)} d t} - {\color{red}{\int{1 d t}}} = - \int{8 \cos{\left(t \right)} d t} - {\color{red}{t}}$$
Aplica la regla del factor constante $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ con $$$c=8$$$ y $$$f{\left(t \right)} = \cos{\left(t \right)}$$$:
$$- t - {\color{red}{\int{8 \cos{\left(t \right)} d t}}} = - t - {\color{red}{\left(8 \int{\cos{\left(t \right)} d t}\right)}}$$
La integral del coseno es $$$\int{\cos{\left(t \right)} d t} = \sin{\left(t \right)}$$$:
$$- t - 8 {\color{red}{\int{\cos{\left(t \right)} d t}}} = - t - 8 {\color{red}{\sin{\left(t \right)}}}$$
Por lo tanto,
$$\int{\left(- 8 \cos{\left(t \right)} - 1\right)d t} = - t - 8 \sin{\left(t \right)}$$
Añade la constante de integración:
$$\int{\left(- 8 \cos{\left(t \right)} - 1\right)d t} = - t - 8 \sin{\left(t \right)}+C$$
Respuesta
$$$\int \left(- 8 \cos{\left(t \right)} - 1\right)\, dt = \left(- t - 8 \sin{\left(t \right)}\right) + C$$$A