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Problem:

The integral to solve is: $\int (3x^2 + 4x - 5) \, dx$

Steps to Solve:

1. Break down the integral: $\int (3x^2 + 4x - 5) \, dx = \int 3x^2 \, dx + \int 4x \, dx - \int 5 \, dx$

2. Solve each term separately:

   • For $\int 3x^2 \, dx$, use the power rule: $\int x^n \, dx = \frac{x^{n+1}}{n+1}$: $\int 3x^2 \, dx = 3 \cdot \frac{x^{2+1}}{2+1} = 3 \cdot \frac{x^3}{3} = x^3$
   • For $\int 4x \, dx$: $\int 4x \, dx = 4 \cdot \frac{x^{1+1}}{1+1} = 4 \cdot \frac{x^2}{2} = 2x^2$
   • For $\int 5 \, dx$: $\int 5 \, dx = 5x$

3. Combine the results: $\int (3x^2 + 4x - 5) \, dx = x^3 + 2x^2 - 5x + C$

Final Answer:

The correct integral is: $y = x^3 + 2x^2 - 5x + C$

Review of the Provided Solution:

From your image, the boxed final answer matches the correct result: $y = x^3 + 2x^2 - 5x + C$

This means the solution is correct.

Do you want me to explain any of the steps in more detail or clarify something specific?

Related Questions:

1. What is the power rule for integration, and why is it used here?
2. How does the constant of integration $C$ come into play when solving indefinite integrals?
3. Could this problem be approached using a definite integral? If so, how would the process differ?
4. How does this solution compare to the derivative process of the resulting function?
5. Why is breaking down the integral into parts (as shown) a useful technique?

Tip:

Always check each term's integration step when solving multi-term integrals to ensure accuracy!