https://file.aiquickdraw.com/mathbot/file/1724064930792vmnqgqor.jpg

The problem in the image appears to be in Slovenian. The task is to find the indefinite integral:

\[
\int \frac{3x+1}{x^2} \, dx
\]

Let's solve this step by step.

First, rewrite the integrand:

\[
\frac{3x+1}{x^2} = \frac{3x}{x^2} + \frac{1}{x^2} = \frac{3}{x} + x^{-2}
\]

Now, split the integral:

\[
\int \left( \frac{3}{x} + x^{-2} \right) \, dx = \int \frac{3}{x} \, dx + \int x^{-2} \, dx
\]

Integrate each term separately:

\[
\int \frac{3}{x} \, dx = 3 \ln |x| + C_1
\]

\[
\int x^{-2} \, dx = \int x^{-2} \, dx = -x^{-1} + C_2 = -\frac{1}{x} + C_2
\]

Combine the results and include the constant of integration \(C\):

\[
3 \ln |x| - \frac{1}{x} + C
\]

So, the final answer is:

\[
\int \frac{3x+1}{x^2} \, dx = 3 \ln |x| - \frac{1}{x} + C
\]

Would you like any details or have any questions?

Here are 5 related questions to expand on this information:

1. How do you handle integration when dealing with polynomials in the denominator?
2. What are some common techniques for integrating rational functions?
3. How do you integrate functions involving logarithms?
4. What are the properties of logarithmic functions that are useful in integration?
5. How do you determine the constant of integration when given initial conditions?

**Tip:** When dealing with integrals involving rational functions, it can be helpful to simplify the integrand first, such as by using algebraic manipulation to split fractions or rewrite terms in a more integrable form.

Explore the step-by-step solution to finding the indefinite integral of (3x+1)/(x^2). Learn how to integrate rational functions and handle terms involving logarithms. Suitable for advanced high school students and anyone studying calculus.

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