Integral of $$$x^{5} \ln\left(7 x\right)$$$

Find $$$\int x^{5} \ln\left(7 x\right)\, dx$$$.

For the integral $$$\int{x^{5} \ln{\left(7 x \right)} d x}$$$, use integration by parts $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Let $$$\operatorname{u}=\ln{\left(7 x \right)}$$$ and $$$\operatorname{dv}=x^{5} dx$$$.

Then $$$\operatorname{du}=\left(\ln{\left(7 x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (steps can be seen ») and $$$\operatorname{v}=\int{x^{5} d x}=\frac{x^{6}}{6}$$$ (steps can be seen »).

Thus,

$${\color{red}{\int{x^{5} \ln{\left(7 x \right)} d x}}}={\color{red}{\left(\ln{\left(7 x \right)} \cdot \frac{x^{6}}{6}-\int{\frac{x^{6}}{6} \cdot \frac{1}{x} d x}\right)}}={\color{red}{\left(\frac{x^{6} \ln{\left(7 x \right)}}{6} - \int{\frac{x^{5}}{6} d x}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{6}$$$ and $$$f{\left(x \right)} = x^{5}$$$:

$$\frac{x^{6} \ln{\left(7 x \right)}}{6} - {\color{red}{\int{\frac{x^{5}}{6} d x}}} = \frac{x^{6} \ln{\left(7 x \right)}}{6} - {\color{red}{\left(\frac{\int{x^{5} d x}}{6}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=5$$$:

$$\frac{x^{6} \ln{\left(7 x \right)}}{6} - \frac{{\color{red}{\int{x^{5} d x}}}}{6}=\frac{x^{6} \ln{\left(7 x \right)}}{6} - \frac{{\color{red}{\frac{x^{1 + 5}}{1 + 5}}}}{6}=\frac{x^{6} \ln{\left(7 x \right)}}{6} - \frac{{\color{red}{\left(\frac{x^{6}}{6}\right)}}}{6}$$

Therefore,

$$\int{x^{5} \ln{\left(7 x \right)} d x} = \frac{x^{6} \ln{\left(7 x \right)}}{6} - \frac{x^{6}}{36}$$

Simplify:

$$\int{x^{5} \ln{\left(7 x \right)} d x} = \frac{x^{6} \left(6 \ln{\left(x \right)} - 1 + 6 \ln{\left(7 \right)}\right)}{36}$$

Add the constant of integration:

$$\int{x^{5} \ln{\left(7 x \right)} d x} = \frac{x^{6} \left(6 \ln{\left(x \right)} - 1 + 6 \ln{\left(7 \right)}\right)}{36}+C$$

$$$\int x^{5} \ln\left(7 x\right)\, dx = \frac{x^{6} \left(6 \ln\left(x\right) - 1 + 6 \ln\left(7\right)\right)}{36} + C$$$A