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Integral de $$$\frac{\ln\left(x\right)}{x^{2}}$$$
Calculadora relacionada: Calculadora de integrales definidas e impropias
Tu entrada
Halla $$$\int \frac{\ln\left(x\right)}{x^{2}}\, dx$$$.
Solución
Para la integral $$$\int{\frac{\ln{\left(x \right)}}{x^{2}} d x}$$$, utiliza la integración por partes $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.
Sean $$$\operatorname{u}=\ln{\left(x \right)}$$$ y $$$\operatorname{dv}=\frac{dx}{x^{2}}$$$.
Entonces $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (los pasos pueden verse ») y $$$\operatorname{v}=\int{\frac{1}{x^{2}} d x}=- \frac{1}{x}$$$ (los pasos pueden verse »).
Por lo tanto,
$${\color{red}{\int{\frac{\ln{\left(x \right)}}{x^{2}} d x}}}={\color{red}{\left(\ln{\left(x \right)} \cdot \left(- \frac{1}{x}\right)-\int{\left(- \frac{1}{x}\right) \cdot \frac{1}{x} d x}\right)}}={\color{red}{\left(- \int{\left(- \frac{1}{x^{2}}\right)d x} - \frac{\ln{\left(x \right)}}{x}\right)}}$$
Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=-1$$$ y $$$f{\left(x \right)} = \frac{1}{x^{2}}$$$:
$$- {\color{red}{\int{\left(- \frac{1}{x^{2}}\right)d x}}} - \frac{\ln{\left(x \right)}}{x} = - {\color{red}{\left(- \int{\frac{1}{x^{2}} d x}\right)}} - \frac{\ln{\left(x \right)}}{x}$$
Aplica la regla de la potencia $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=-2$$$:
$${\color{red}{\int{\frac{1}{x^{2}} d x}}} - \frac{\ln{\left(x \right)}}{x}={\color{red}{\int{x^{-2} d x}}} - \frac{\ln{\left(x \right)}}{x}={\color{red}{\frac{x^{-2 + 1}}{-2 + 1}}} - \frac{\ln{\left(x \right)}}{x}={\color{red}{\left(- x^{-1}\right)}} - \frac{\ln{\left(x \right)}}{x}={\color{red}{\left(- \frac{1}{x}\right)}} - \frac{\ln{\left(x \right)}}{x}$$
Por lo tanto,
$$\int{\frac{\ln{\left(x \right)}}{x^{2}} d x} = - \frac{\ln{\left(x \right)}}{x} - \frac{1}{x}$$
Simplificar:
$$\int{\frac{\ln{\left(x \right)}}{x^{2}} d x} = \frac{- \ln{\left(x \right)} - 1}{x}$$
Añade la constante de integración:
$$\int{\frac{\ln{\left(x \right)}}{x^{2}} d x} = \frac{- \ln{\left(x \right)} - 1}{x}+C$$
Respuesta
$$$\int \frac{\ln\left(x\right)}{x^{2}}\, dx = \frac{- \ln\left(x\right) - 1}{x} + C$$$A