To find the antiderivative of $f(x) = 8x^{10} + 9x^7 - 6x^2 - 1$, we integrate each term of $f(x)$ with respect to $x$.

Solution:

1. Integrate each term individually: $\int 8x^{10} \, dx = \frac{8}{11} x^{11}$ $\int 9x^7 \, dx = \frac{9}{8} x^8$ $\int -6x^2 \, dx = -2x^3$ $\int -1 \, dx = -x$

2. Combine the results and add the constant of integration $C$: $\int f(x) \, dx = \frac{8}{11} x^{11} + \frac{9}{8} x^8 - 2x^3 - x + C$

Final Answer:

$F(x) = \frac{8}{11} x^{11} + \frac{9}{8} x^8 - 2x^3 - x + C$

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Related Questions:

1. How do you determine the antiderivative of a power function?
2. What is the constant of integration, and why is it added in indefinite integrals?
3. How do you apply the power rule in integration?
4. Can you explain how to integrate negative powers or fractional exponents?
5. What are the differences between definite and indefinite integrals?

Tip:

Remember, for an expression $ax^n$, the antiderivative is $\frac{a}{n+1} x^{n+1}$ when $n \neq -1$.