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The given problem asks to find the function \( g(x) \) whose derivative is \( g'(x) = 3x^2 - 5 \).

To find \( g(x) \), we integrate the derivative:

\[
g(x) = \int (3x^2 - 5) \, dx
\]

Integrating term by term:

\[
\int 3x^2 \, dx = 3 \times \frac{x^3}{3} = x^3
\]
\[
\int -5 \, dx = -5x
\]

Thus, the function \( g(x) \) is:

\[
g(x) = x^3 - 5x + C
\]

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

Would you like more details on any steps or have other questions related to this topic? Here are some follow-up questions you might consider:

1. How would you find the value of the constant \( C \) if you were given an initial condition?
2. Can you explain how to check if this solution is correct by differentiating \( g(x) \)?
3. What would the graph of \( g(x) \) look like based on this equation?
4. How does the derivative \( g'(x) \) relate to the shape of the graph of \( g(x) \)?
5. What are some real-world applications of using integration to find a function given its derivative?

**Tip:** Remember, the constant of integration \( C \) can only be determined if additional information, such as a point through which the function passes, is provided.

Learn how to find the function g(x) by integrating its derivative g'(x) = 3x^2 - 5. This step-by-step solution involves calculus concepts such as integration and the constant of integration. Suitable for high school students in Grades 11-12.

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