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Integral de $$$\cot{\left(2 x \right)}$$$
Calculadora relacionada: Calculadora de integrales definidas e impropias
Tu entrada
Halla $$$\int \cot{\left(2 x \right)}\, dx$$$.
Solución
Sea $$$u=2 x$$$.
Entonces $$$du=\left(2 x\right)^{\prime }dx = 2 dx$$$ (los pasos pueden verse »), y obtenemos que $$$dx = \frac{du}{2}$$$.
Por lo tanto,
$${\color{red}{\int{\cot{\left(2 x \right)} d x}}} = {\color{red}{\int{\frac{\cot{\left(u \right)}}{2} d u}}}$$
Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{1}{2}$$$ y $$$f{\left(u \right)} = \cot{\left(u \right)}$$$:
$${\color{red}{\int{\frac{\cot{\left(u \right)}}{2} d u}}} = {\color{red}{\left(\frac{\int{\cot{\left(u \right)} d u}}{2}\right)}}$$
Reescribe la cotangente como $$$\cot\left( u \right)=\frac{\cos\left( u \right)}{\sin\left( u \right)}$$$:
$$\frac{{\color{red}{\int{\cot{\left(u \right)} d u}}}}{2} = \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{\sin{\left(u \right)}} d u}}}}{2}$$
Sea $$$v=\sin{\left(u \right)}$$$.
Entonces $$$dv=\left(\sin{\left(u \right)}\right)^{\prime }du = \cos{\left(u \right)} du$$$ (los pasos pueden verse »), y obtenemos que $$$\cos{\left(u \right)} du = dv$$$.
La integral se convierte en
$$\frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{\sin{\left(u \right)}} d u}}}}{2} = \frac{{\color{red}{\int{\frac{1}{v} d v}}}}{2}$$
La integral de $$$\frac{1}{v}$$$ es $$$\int{\frac{1}{v} d v} = \ln{\left(\left|{v}\right| \right)}$$$:
$$\frac{{\color{red}{\int{\frac{1}{v} d v}}}}{2} = \frac{{\color{red}{\ln{\left(\left|{v}\right| \right)}}}}{2}$$
Recordemos que $$$v=\sin{\left(u \right)}$$$:
$$\frac{\ln{\left(\left|{{\color{red}{v}}}\right| \right)}}{2} = \frac{\ln{\left(\left|{{\color{red}{\sin{\left(u \right)}}}}\right| \right)}}{2}$$
Recordemos que $$$u=2 x$$$:
$$\frac{\ln{\left(\left|{\sin{\left({\color{red}{u}} \right)}}\right| \right)}}{2} = \frac{\ln{\left(\left|{\sin{\left({\color{red}{\left(2 x\right)}} \right)}}\right| \right)}}{2}$$
Por lo tanto,
$$\int{\cot{\left(2 x \right)} d x} = \frac{\ln{\left(\left|{\sin{\left(2 x \right)}}\right| \right)}}{2}$$
Añade la constante de integración:
$$\int{\cot{\left(2 x \right)} d x} = \frac{\ln{\left(\left|{\sin{\left(2 x \right)}}\right| \right)}}{2}+C$$
Respuesta
$$$\int \cot{\left(2 x \right)}\, dx = \frac{\ln\left(\left|{\sin{\left(2 x \right)}}\right|\right)}{2} + C$$$A