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

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

A entrada é reescrita como: $$$\int{\ln{\left(x \sqrt{x^{21}} \right)} d x}=\int{\frac{23 \ln{\left(x \right)}}{2} d x}$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=\frac{23}{2}$$$ e $$$f{\left(x \right)} = \ln{\left(x \right)}$$$:

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

Para a integral $$$\int{\ln{\left(x \right)} d x}$$$, use integração por partes $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Sejam $$$\operatorname{u}=\ln{\left(x \right)}$$$ e $$$\operatorname{dv}=dx$$$.

Então $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (os passos podem ser vistos ») e $$$\operatorname{v}=\int{1 d x}=x$$$ (os passos podem ser vistos »).

A integral pode ser reescrita como

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

Aplique a regra da constante $$$\int c\, dx = c x$$$ usando $$$c=1$$$:

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

Portanto,

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

Simplifique:

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

Adicione a constante de integração:

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

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