Answer :
To find the antiderivative of the function [tex]\( f(x) = 9x^9 + 7x^7 - 3x^4 - 3 \)[/tex], we'll integrate each term with respect to [tex]\( x \)[/tex].
1. Integrate [tex]\( 9x^9 \)[/tex]:
- The rule for integrating [tex]\( x^n \)[/tex] is to increase the exponent by 1 and divide by the new exponent:
[tex]\[
\int 9x^9 \, dx = \frac{9}{10} x^{10}
\][/tex]
2. Integrate [tex]\( 7x^7 \)[/tex]:
- Apply the same rule:
[tex]\[
\int 7x^7 \, dx = \frac{7}{8} x^8
\][/tex]
3. Integrate [tex]\(-3x^4\)[/tex]:
- Again, using the power rule:
[tex]\[
\int -3x^4 \, dx = -\frac{3}{5} x^5
\][/tex]
4. Integrate [tex]\(-3\)[/tex]:
- The integral of a constant [tex]\( a \)[/tex] is simply [tex]\( ax \)[/tex]:
[tex]\[
\int -3 \, dx = -3x
\][/tex]
Now, combine all these results to write the complete antiderivative:
[tex]\[
F(x) = \frac{9}{10} x^{10} + \frac{7}{8} x^8 - \frac{3}{5} x^5 - 3x + C
\][/tex]
where [tex]\( C \)[/tex] is the constant of integration.
1. Integrate [tex]\( 9x^9 \)[/tex]:
- The rule for integrating [tex]\( x^n \)[/tex] is to increase the exponent by 1 and divide by the new exponent:
[tex]\[
\int 9x^9 \, dx = \frac{9}{10} x^{10}
\][/tex]
2. Integrate [tex]\( 7x^7 \)[/tex]:
- Apply the same rule:
[tex]\[
\int 7x^7 \, dx = \frac{7}{8} x^8
\][/tex]
3. Integrate [tex]\(-3x^4\)[/tex]:
- Again, using the power rule:
[tex]\[
\int -3x^4 \, dx = -\frac{3}{5} x^5
\][/tex]
4. Integrate [tex]\(-3\)[/tex]:
- The integral of a constant [tex]\( a \)[/tex] is simply [tex]\( ax \)[/tex]:
[tex]\[
\int -3 \, dx = -3x
\][/tex]
Now, combine all these results to write the complete antiderivative:
[tex]\[
F(x) = \frac{9}{10} x^{10} + \frac{7}{8} x^8 - \frac{3}{5} x^5 - 3x + C
\][/tex]
where [tex]\( C \)[/tex] is the constant of integration.
