Integral of $$$\frac{9}{10 x - 20}$$$

Find $$$\int \frac{9}{10 x - 20}\, dx$$$.

Simplify the integrand:

$${\color{red}{\int{\frac{9}{10 x - 20} d x}}} = {\color{red}{\int{\frac{9}{10 \left(x - 2\right)} d x}}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{9}{10}$$$ and $$$f{\left(x \right)} = \frac{1}{x - 2}$$$:

$${\color{red}{\int{\frac{9}{10 \left(x - 2\right)} d x}}} = {\color{red}{\left(\frac{9 \int{\frac{1}{x - 2} d x}}{10}\right)}}$$

Let $$$u=x - 2$$$.

Then $$$du=\left(x - 2\right)^{\prime }dx = 1 dx$$$ (steps can be seen »), and we have that $$$dx = du$$$.

The integral can be rewritten as

$$\frac{9 {\color{red}{\int{\frac{1}{x - 2} d x}}}}{10} = \frac{9 {\color{red}{\int{\frac{1}{u} d u}}}}{10}$$

The integral of $$$\frac{1}{u}$$$ is $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

$$\frac{9 {\color{red}{\int{\frac{1}{u} d u}}}}{10} = \frac{9 {\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{10}$$

Recall that $$$u=x - 2$$$:

$$\frac{9 \ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{10} = \frac{9 \ln{\left(\left|{{\color{red}{\left(x - 2\right)}}}\right| \right)}}{10}$$

Therefore,

$$\int{\frac{9}{10 x - 20} d x} = \frac{9 \ln{\left(\left|{x - 2}\right| \right)}}{10}$$

Add the constant of integration:

$$\int{\frac{9}{10 x - 20} d x} = \frac{9 \ln{\left(\left|{x - 2}\right| \right)}}{10}+C$$

$$$\int \frac{9}{10 x - 20}\, dx = \frac{9 \ln\left(\left|{x - 2}\right|\right)}{10} + C$$$A