To solve the integral 

\[
\int \frac{2v - 7 + v^2}{v} \, dv,
\]

we can first simplify the expression by dividing each term in the numerator by \( v \). This gives us:

\[
\int \left( \frac{2v}{v} - \frac{7}{v} + \frac{v^2}{v} \right) dv = \int \left( 2 - \frac{7}{v} + v \right) dv.
\]

Now we can integrate each term separately:

\[
\int 2 \, dv = 2v,
\]

\[
\int -\frac{7}{v} \, dv = -7 \ln|v|,
\]

\[
\int v \, dv = \frac{v^2}{2}.
\]

Thus, the result of the integral is:

\[
2v - 7 \ln|v| + \frac{v^2}{2} + C,
\]

where \( C \) is the constant of integration.

### Final Answer:

\[
\boxed{2v - 7 \ln|v| + \frac{v^2}{2} + C.}
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you integrate rational functions with a quadratic numerator?
2. What are the steps to simplify integrals before solving?
3. How do logarithmic functions appear in integrals?
4. Can you explain how to handle absolute values in logarithms during integration?
5. What are common techniques for integrating polynomial terms?

### Tip:
When integrating functions with multiple terms, always check if simplifying the expression first can make the process easier.

Learn how to solve the integral ∫ (2v - 7 + v^2) / v dv with a detailed step-by-step solution. This problem covers key calculus concepts such as polynomial division, logarithmic integration, and applying the power rule. Suitable for college students studying Calculus I.

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