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]
[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]
