What is the integral 1/x ?
$$\small{\text{
We substitute : $ x = e^u \qquad \ dx = e^u\ du
$
}}\\
\small{\text{
$\int {\frac{1}{x} }\ dx = \int{\frac{1}{e^u} }\cdot e^u \ du = \int{1}\ du = u
$
}}\\\\
\small{\text{Back substitute :
$\int {\frac{1}{x} }\ dx = u \qquad u = \ln{(x)}
$
}}\\
\small{\text{
$\int {\frac{1}{x} }\ dx = \ln{(x)} + c
$
}}$$
What is the integral 1/x ?
$$\small{\text{
We substitute : $ x = e^u \qquad \ dx = e^u\ du
$
}}\\
\small{\text{
$\int {\frac{1}{x} }\ dx = \int{\frac{1}{e^u} }\cdot e^u \ du = \int{1}\ du = u
$
}}\\\\
\small{\text{Back substitute :
$\int {\frac{1}{x} }\ dx = u \qquad u = \ln{(x)}
$
}}\\
\small{\text{
$\int {\frac{1}{x} }\ dx = \ln{(x)} + c
$
}}$$