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Integral de $$$\frac{\tan{\left(y \right)}}{\ln\left(\cos{\left(y \right)}\right)}$$$
Calculadora relacionada: Calculadora de integrales definidas e impropias
Tu entrada
Halla $$$\int \frac{\tan{\left(y \right)}}{\ln\left(\cos{\left(y \right)}\right)}\, dy$$$.
Solución
Sea $$$u=\cos{\left(y \right)}$$$.
Entonces $$$du=\left(\cos{\left(y \right)}\right)^{\prime }dy = - \sin{\left(y \right)} dy$$$ (los pasos pueden verse »), y obtenemos que $$$\sin{\left(y \right)} dy = - du$$$.
Entonces,
$${\color{red}{\int{\frac{\tan{\left(y \right)}}{\ln{\left(\cos{\left(y \right)} \right)}} d y}}} = {\color{red}{\int{\left(- \frac{1}{u \ln{\left(u \right)}}\right)d u}}}$$
Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=-1$$$ y $$$f{\left(u \right)} = \frac{1}{u \ln{\left(u \right)}}$$$:
$${\color{red}{\int{\left(- \frac{1}{u \ln{\left(u \right)}}\right)d u}}} = {\color{red}{\left(- \int{\frac{1}{u \ln{\left(u \right)}} d u}\right)}}$$
Sea $$$v=\ln{\left(u \right)}$$$.
Entonces $$$dv=\left(\ln{\left(u \right)}\right)^{\prime }du = \frac{du}{u}$$$ (los pasos pueden verse »), y obtenemos que $$$\frac{du}{u} = dv$$$.
Por lo tanto,
$$- {\color{red}{\int{\frac{1}{u \ln{\left(u \right)}} d u}}} = - {\color{red}{\int{\frac{1}{v} d v}}}$$
La integral de $$$\frac{1}{v}$$$ es $$$\int{\frac{1}{v} d v} = \ln{\left(\left|{v}\right| \right)}$$$:
$$- {\color{red}{\int{\frac{1}{v} d v}}} = - {\color{red}{\ln{\left(\left|{v}\right| \right)}}}$$
Recordemos que $$$v=\ln{\left(u \right)}$$$:
$$- \ln{\left(\left|{{\color{red}{v}}}\right| \right)} = - \ln{\left(\left|{{\color{red}{\ln{\left(u \right)}}}}\right| \right)}$$
Recordemos que $$$u=\cos{\left(y \right)}$$$:
$$- \ln{\left(\left|{\ln{\left({\color{red}{u}} \right)}}\right| \right)} = - \ln{\left(\left|{\ln{\left({\color{red}{\cos{\left(y \right)}}} \right)}}\right| \right)}$$
Por lo tanto,
$$\int{\frac{\tan{\left(y \right)}}{\ln{\left(\cos{\left(y \right)} \right)}} d y} = - \ln{\left(\left|{\ln{\left(\cos{\left(y \right)} \right)}}\right| \right)}$$
Añade la constante de integración:
$$\int{\frac{\tan{\left(y \right)}}{\ln{\left(\cos{\left(y \right)} \right)}} d y} = - \ln{\left(\left|{\ln{\left(\cos{\left(y \right)} \right)}}\right| \right)}+C$$
Respuesta
$$$\int \frac{\tan{\left(y \right)}}{\ln\left(\cos{\left(y \right)}\right)}\, dy = - \ln\left(\left|{\ln\left(\cos{\left(y \right)}\right)}\right|\right) + C$$$A