Answer :
To find an antiderivative of
[tex]$$
f(x)=3x^9+2x^7-9x^2-9,
$$[/tex]
we integrate each term of the function separately.
1. For the first term, integrate
[tex]$$
3x^9:
$$[/tex]
[tex]$$
\int 3x^9 \, dx = 3 \cdot \frac{x^{10}}{10} = \frac{3}{10}x^{10}.
$$[/tex]
2. For the second term, integrate
[tex]$$
2x^7:
$$[/tex]
[tex]$$
\int 2x^7 \, dx = 2 \cdot \frac{x^8}{8} = \frac{1}{4}x^8.
$$[/tex]
3. For the third term, integrate
[tex]$$
-9x^2:
$$[/tex]
[tex]$$
\int -9x^2 \, dx = -9 \cdot \frac{x^3}{3} = -3x^3.
$$[/tex]
4. For the constant term, integrate
[tex]$$
-9:
$$[/tex]
[tex]$$
\int -9 \, dx = -9x.
$$[/tex]
Now, combining all these results gives the antiderivative:
[tex]$$
F(x)=\frac{3}{10}x^{10}+\frac{1}{4}x^8-3x^3-9x+C,
$$[/tex]
where [tex]$C$[/tex] is the constant of integration.
Thus, one antiderivative of [tex]$f(x)$[/tex] is
[tex]$$
\frac{3}{10}x^{10}+\frac{1}{4}x^8-3x^3-9x.
$$[/tex]
[tex]$$
f(x)=3x^9+2x^7-9x^2-9,
$$[/tex]
we integrate each term of the function separately.
1. For the first term, integrate
[tex]$$
3x^9:
$$[/tex]
[tex]$$
\int 3x^9 \, dx = 3 \cdot \frac{x^{10}}{10} = \frac{3}{10}x^{10}.
$$[/tex]
2. For the second term, integrate
[tex]$$
2x^7:
$$[/tex]
[tex]$$
\int 2x^7 \, dx = 2 \cdot \frac{x^8}{8} = \frac{1}{4}x^8.
$$[/tex]
3. For the third term, integrate
[tex]$$
-9x^2:
$$[/tex]
[tex]$$
\int -9x^2 \, dx = -9 \cdot \frac{x^3}{3} = -3x^3.
$$[/tex]
4. For the constant term, integrate
[tex]$$
-9:
$$[/tex]
[tex]$$
\int -9 \, dx = -9x.
$$[/tex]
Now, combining all these results gives the antiderivative:
[tex]$$
F(x)=\frac{3}{10}x^{10}+\frac{1}{4}x^8-3x^3-9x+C,
$$[/tex]
where [tex]$C$[/tex] is the constant of integration.
Thus, one antiderivative of [tex]$f(x)$[/tex] is
[tex]$$
\frac{3}{10}x^{10}+\frac{1}{4}x^8-3x^3-9x.
$$[/tex]
