Integral de $$$\ln\left(x \sqrt{x^{21}}\right)$$$

Halla $$$\int \ln\left(x \sqrt{x^{21}}\right)\, dx$$$.

La entrada se reescribe: $$$\int{\ln{\left(x \sqrt{x^{21}} \right)} d x}=\int{\frac{23 \ln{\left(x \right)}}{2} d x}$$$.

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=\frac{23}{2}$$$ y $$$f{\left(x \right)} = \ln{\left(x \right)}$$$:

$${\color{red}{\int{\frac{23 \ln{\left(x \right)}}{2} d x}}} = {\color{red}{\left(\frac{23 \int{\ln{\left(x \right)} d x}}{2}\right)}}$$

Para la integral $$$\int{\ln{\left(x \right)} d x}$$$, 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 $$$\operatorname{dv}=dx$$$.

Entonces $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (los pasos pueden verse ») y $$$\operatorname{v}=\int{1 d x}=x$$$ (los pasos pueden verse »).

Por lo tanto,

$$\frac{23 {\color{red}{\int{\ln{\left(x \right)} d x}}}}{2}=\frac{23 {\color{red}{\left(\ln{\left(x \right)} \cdot x-\int{x \cdot \frac{1}{x} d x}\right)}}}{2}=\frac{23 {\color{red}{\left(x \ln{\left(x \right)} - \int{1 d x}\right)}}}{2}$$

Aplica la regla de la constante $$$\int c\, dx = c x$$$ con $$$c=1$$$:

$$\frac{23 x \ln{\left(x \right)}}{2} - \frac{23 {\color{red}{\int{1 d x}}}}{2} = \frac{23 x \ln{\left(x \right)}}{2} - \frac{23 {\color{red}{x}}}{2}$$

Por lo tanto,

$$\int{\frac{23 \ln{\left(x \right)}}{2} d x} = \frac{23 x \ln{\left(x \right)}}{2} - \frac{23 x}{2}$$

Simplificar:

$$\int{\frac{23 \ln{\left(x \right)}}{2} d x} = \frac{23 x \left(\ln{\left(x \right)} - 1\right)}{2}$$

Añade la constante de integración:

$$\int{\frac{23 \ln{\left(x \right)}}{2} d x} = \frac{23 x \left(\ln{\left(x \right)} - 1\right)}{2}+C$$

$$$\int \ln\left(x \sqrt{x^{21}}\right)\, dx = \frac{23 x \left(\ln\left(x\right) - 1\right)}{2} + C$$$A