The handwritten solution provided involves logarithmic and exponential functions. Let's break down the steps from the image:

First line:

$2 \log |1 - e^x| + C' = \int (\log | \text{tan(y)} | + C'')$

• The left-hand side involves the logarithm of $1 - e^x$, with a constant $C'$ added.
• The right-hand side represents the integral of $\log | \text{tan}(y) |$ plus a constant $C''$.

Second line:

• It seems to indicate the symbol "∴" (therefore), implying a result based on the previous equation.

Third line:

$\text{tan}(y) = (1 - e^x)^2 + C$

• The solution simplifies to this, with $\text{tan}(y)$ expressed in terms of $1 - e^x$.
• The constant $C' - C'' = C$ is simplified and factored into the final expression, consolidating the constants.

Conclusion:

The process suggests the following steps were taken:

1. Integrating both sides involving the logarithmic and tangent functions.
2. Solving the integral on the right-hand side and simplifying the left side, with both sides being equated.
3. The logarithmic and exponential expressions lead to the final result for $\text{tan}(y)$, with an additional constant term $C$.

Would you like further details on any specific part of this solution?

Here are five related questions:

1. How do you integrate logarithmic functions involving constants like $C'$ and $C''$?
2. Why is the constant of integration simplified as $C' - C'' = C$?
3. Can you explain the relationship between $\log |1 - e^x|$ and its integral?
4. What techniques are used to simplify expressions involving $\log$ and $\tan$?
5. How does the exponential term $e^x$ affect the solution of the equation?

Tip: When dealing with logarithmic equations, always remember the domain restrictions, especially with expressions inside absolute values.