Integral de $$$\ln\left(x\right)$$$
Calculadora relacionada: Calculadora de integrales definidas e impropias
Tu aportación
Encuentra $$$\int \ln\left(x\right)\, dx$$$.
Solución
For the integral $$$\int{\ln{\left(x \right)} d x}$$$, use integration by parts $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.
Let $$$\operatorname{u}=\ln{\left(x \right)}$$$ and $$$\operatorname{dv}=dx$$$.
Then $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (steps can be seen here) and $$$\operatorname{v}=\int{1 d x}=x$$$ (steps can be seen here).
So,
$${\color{red}{\int{\ln{\left(x \right)} d x}}}={\color{red}{\left(\ln{\left(x \right)} \cdot x-\int{x \cdot \frac{1}{x} d x}\right)}}={\color{red}{\left(x \ln{\left(x \right)} - \int{1 d x}\right)}}$$
Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=1$$$:
$$x \ln{\left(x \right)} - {\color{red}{\int{1 d x}}} = x \ln{\left(x \right)} - {\color{red}{x}}$$
Therefore,
$$\int{\ln{\left(x \right)} d x} = x \ln{\left(x \right)} - x$$
Simplify:
$$\int{\ln{\left(x \right)} d x} = x \left(\ln{\left(x \right)} - 1\right)$$
Add the constant of integration:
$$\int{\ln{\left(x \right)} d x} = x \left(\ln{\left(x \right)} - 1\right)+C$$
Answer: $$$\int{\ln{\left(x \right)} d x}=x \left(\ln{\left(x \right)} - 1\right)+C$$$