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