The image contains three different attempts at solving the integral $\int (x^4 + 3x - 9) \, dx$. Let's analyze and solve it correctly.

Correct Approach:

Given the integral: $\int (x^4 + 3x - 9) \, dx$

We solve it term by term:

1. Integral of $x^4$: $\int x^4 \, dx = \frac{x^5}{5}$

2. Integral of $3x$: $\int 3x \, dx = \frac{3x^2}{2}$

3. Integral of $-9$: $\int -9 \, dx = -9x$

Now, combining all the results: $\int (x^4 + 3x - 9) \, dx = \frac{x^5}{5} + \frac{3x^2}{2} - 9x + C$

Verification:

• First equation: $\frac{x^5}{5} + \frac{3x^2}{2} - 9x + C$ — Correct.
• Second equation: $\frac{x^5}{5} + \frac{3x^2}{2} + C - 9$ — Incorrect as the constant $-9$ should be $-9x$.
• Third equation: $\frac{x^5}{5} + C + 3x - 9$ — Incorrect as it omits the integration of $3x$ properly.

Would you like further details or explanations on any specific part of this integration?

Related Questions:

1. How do we apply the power rule in integration?
2. What happens if we have definite bounds in this integral?
3. How do constants affect the result of integration?
4. How can we verify the correctness of an indefinite integral?
5. What are common mistakes to avoid when integrating polynomials?

Tip: Always double-check each term's integration, particularly constants, to avoid common mistakes.