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Consider the function [tex]f(x) = 7x^8 + 10x^6 - 8x^2 - 7[/tex].

Enter the antiderivative of [tex]f(x)[/tex]: [tex]\square + C[/tex].

Answer :

We start with the function

[tex]$$
f(x) = 7x^8 + 10x^6 - 8x^2 - 7.
$$[/tex]

We will find the antiderivative (indefinite integral) term by term.

1. For the first term, compute

[tex]$$
\int 7x^8\,dx.
$$[/tex]

Using the rule

[tex]$$
\int x^n\,dx = \frac{x^{n+1}}{n+1} + C,
$$[/tex]

we have

[tex]$$
\int 7x^8\,dx = 7 \cdot \frac{x^{9}}{9} = \frac{7}{9} x^9.
$$[/tex]

2. For the second term, compute

[tex]$$
\int 10x^6\,dx = 10 \cdot \frac{x^{7}}{7} = \frac{10}{7} x^7.
$$[/tex]

3. For the third term, compute

[tex]$$
\int (-8x^2)\,dx = -8 \cdot \frac{x^{3}}{3} = -\frac{8}{3} x^3.
$$[/tex]

4. For the fourth term, compute

[tex]$$
\int (-7)\,dx = -7x.
$$[/tex]

Now, putting all these results together, the antiderivative of [tex]$f(x)$[/tex] is

[tex]$$
\frac{7}{9} x^9 + \frac{10}{7} x^7 - \frac{8}{3} x^3 - 7x + C,
$$[/tex]

where [tex]$C$[/tex] is the constant of integration.

Thus, the final answer is

[tex]$$
\boxed{\frac{7}{9} x^9 + \frac{10}{7} x^7 - \frac{8}{3} x^3 - 7x + C.}
$$[/tex]

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