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To solve the integral \(\int \left(9x^8 - \frac{1}{3}x^7\right) dx\), we will integrate each term separately.

1. Integrate \(9x^8\):
\[
\int 9x^8 \, dx = \frac{9}{8+1} x^{8+1} = \frac{9}{9} x^9 = x^9
\]

2. Integrate \(-\frac{1}{3}x^7\):
\[
\int -\frac{1}{3}x^7 \, dx = -\frac{1}{3} \cdot \frac{1}{7+1} x^{7+1} = -\frac{1}{3} \cdot \frac{1}{8} x^8 = -\frac{1}{24} x^8
\]

Combining these results:
\[
\int \left(9x^8 - \frac{1}{3}x^7\right) dx = x^9 - \frac{1}{24} x^8 + C
\]

None of the options in the given choices match this form directly. However, since the constants may appear differently in simplified form, let’s closely match our result with one of the options given:

Comparing with the options:
- Option A: \(\frac{9}{8}x^8 + \frac{4}{3}x^{7/3} + C\)
- Option B: \(\frac{9}{8}x^8 - \frac{1}{3}x^{7/3} + C\)
- Option C: \(9x^8 - x^{4/3} + C\)
- Option D: \(\frac{9}{8}x^8 - \frac{1}{3}x^8 + C\)

The form \(x^9 - \frac{1}{24} x^8 + C\) does not match any of the provided answer choices directly. 

Please provide more context or details if there’s any specific instruction or if there's a potential error in the problem statement. Let me know if you need further details or have any questions.

Learn how to solve the integral of the polynomial function 9x^8 - (1/3)x^7 step-by-step with detailed instructions. This problem involves integrating polynomial functions, applying the power rule, and identifying constant terms. Suitable for college-level students studying calculus.

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