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Integral de $$$\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$$$
Calculadora relacionada: Calculadora de integrales definidas e impropias
Tu entrada
Halla $$$\int \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx$$$.
Solución
Sea $$$u=\sin{\left(x \right)}$$$.
Entonces $$$du=\left(\sin{\left(x \right)}\right)^{\prime }dx = \cos{\left(x \right)} dx$$$ (los pasos pueden verse »), y obtenemos que $$$\cos{\left(x \right)} dx = du$$$.
La integral puede reescribirse como
$${\color{red}{\int{\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} d x}}} = {\color{red}{\int{\frac{1}{u} d u}}}$$
La integral de $$$\frac{1}{u}$$$ es $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:
$${\color{red}{\int{\frac{1}{u} d u}}} = {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$
Recordemos que $$$u=\sin{\left(x \right)}$$$:
$$\ln{\left(\left|{{\color{red}{u}}}\right| \right)} = \ln{\left(\left|{{\color{red}{\sin{\left(x \right)}}}}\right| \right)}$$
Por lo tanto,
$$\int{\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} d x} = \ln{\left(\left|{\sin{\left(x \right)}}\right| \right)}$$
Añade la constante de integración:
$$\int{\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} d x} = \ln{\left(\left|{\sin{\left(x \right)}}\right| \right)}+C$$
Respuesta
$$$\int \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx = \ln\left(\left|{\sin{\left(x \right)}}\right|\right) + C$$$A