Integral de $$$\csc^{2}{\left(x \right)} + 1$$$
Calculadora relacionada: Calculadora de integrales definidas e impropias
Tu entrada
Halla $$$\int \left(\csc^{2}{\left(x \right)} + 1\right)\, dx$$$.
Solución
Integra término a término:
$${\color{red}{\int{\left(\csc^{2}{\left(x \right)} + 1\right)d x}}} = {\color{red}{\left(\int{1 d x} + \int{\csc^{2}{\left(x \right)} d x}\right)}}$$
Aplica la regla de la constante $$$\int c\, dx = c x$$$ con $$$c=1$$$:
$$\int{\csc^{2}{\left(x \right)} d x} + {\color{red}{\int{1 d x}}} = \int{\csc^{2}{\left(x \right)} d x} + {\color{red}{x}}$$
La integral de $$$\csc^{2}{\left(x \right)}$$$ es $$$\int{\csc^{2}{\left(x \right)} d x} = - \cot{\left(x \right)}$$$:
$$x + {\color{red}{\int{\csc^{2}{\left(x \right)} d x}}} = x + {\color{red}{\left(- \cot{\left(x \right)}\right)}}$$
Por lo tanto,
$$\int{\left(\csc^{2}{\left(x \right)} + 1\right)d x} = x - \cot{\left(x \right)}$$
Añade la constante de integración:
$$\int{\left(\csc^{2}{\left(x \right)} + 1\right)d x} = x - \cot{\left(x \right)}+C$$
Respuesta
$$$\int \left(\csc^{2}{\left(x \right)} + 1\right)\, dx = \left(x - \cot{\left(x \right)}\right) + C$$$A