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