Integral de $$$\frac{3}{x - 4}$$$
Calculadora relacionada: Calculadora de integrales definidas e impropias
Tu entrada
Halla $$$\int \frac{3}{x - 4}\, dx$$$.
Solución
Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=3$$$ y $$$f{\left(x \right)} = \frac{1}{x - 4}$$$:
$${\color{red}{\int{\frac{3}{x - 4} d x}}} = {\color{red}{\left(3 \int{\frac{1}{x - 4} d x}\right)}}$$
Sea $$$u=x - 4$$$.
Entonces $$$du=\left(x - 4\right)^{\prime }dx = 1 dx$$$ (los pasos pueden verse »), y obtenemos que $$$dx = du$$$.
La integral puede reescribirse como
$$3 {\color{red}{\int{\frac{1}{x - 4} d x}}} = 3 {\color{red}{\int{\frac{1}{u} d u}}}$$
La integral de $$$\frac{1}{u}$$$ es $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:
$$3 {\color{red}{\int{\frac{1}{u} d u}}} = 3 {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$
Recordemos que $$$u=x - 4$$$:
$$3 \ln{\left(\left|{{\color{red}{u}}}\right| \right)} = 3 \ln{\left(\left|{{\color{red}{\left(x - 4\right)}}}\right| \right)}$$
Por lo tanto,
$$\int{\frac{3}{x - 4} d x} = 3 \ln{\left(\left|{x - 4}\right| \right)}$$
Añade la constante de integración:
$$\int{\frac{3}{x - 4} d x} = 3 \ln{\left(\left|{x - 4}\right| \right)}+C$$
Respuesta
$$$\int \frac{3}{x - 4}\, dx = 3 \ln\left(\left|{x - 4}\right|\right) + C$$$A