https://file.aiquickdraw.com/mathbot/file/1724870367469zqhbrtsg.jpg

The image shows the following equation involving two integrals:

\[
\int \frac{1 - v}{v^2} dv = \int \frac{1}{x} dx
\]

Let's solve both integrals.

### 1. Left-hand Side: \(\int \frac{1 - v}{v^2} dv\)
We can split this integral into two separate terms:

\[
\int \frac{1 - v}{v^2} dv = \int \frac{1}{v^2} dv - \int \frac{v}{v^2} dv
\]

#### Solving \(\int \frac{1}{v^2} dv\):
\[
\int v^{-2} dv = \frac{v^{-1}}{-1} = -\frac{1}{v}
\]

#### Solving \(\int \frac{v}{v^2} dv\):
\[
\int v^{-1} dv = \ln|v|
\]

Thus, the left-hand side becomes:
\[
-\frac{1}{v} - \ln|v|
\]

### 2. Right-hand Side: \(\int \frac{1}{x} dx\)
This is a standard integral:
\[
\int \frac{1}{x} dx = \ln|x|
\]

### Final Equation:
Thus, the equation becomes:
\[
-\frac{1}{v} - \ln|v| = \ln|x| + C
\]
where \(C\) is the constant of integration.

If you'd like further clarification or additional steps, let me know!

### Follow-up Questions:
1. Can you verify the conditions for when \(v\) and \(x\) are positive or negative?
2. How would the solution change if we integrate with respect to definite limits?
3. Can you substitute specific values for \(v\) and \(x\) to check the solution?
4. What is the physical interpretation of this equation, if it comes from a real-world context?
5. How would you find \(v\) as a function of \(x\) from the resulting equation?

### Tip:
Always remember to include the constant of integration when solving indefinite integrals.

Learn how to solve the integral equation involving logarithmic functions step-by-step. This problem covers integration techniques, including the integral of fractions and logarithmic functions. Suitable for advanced high school students and above.

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