Answer :
To find the antiderivative of the function [tex]\( f(x) = 6x^8 + 7x^6 - 8x^2 - 7 \)[/tex], we need to integrate each term of the polynomial individually. Let's do this step-by-step:
1. Integrate each term separately:
- For the term [tex]\( 6x^8 \)[/tex], the antiderivative is:
[tex]\[
\frac{6}{9}x^9 = \frac{2}{3}x^9
\][/tex]
Here, we add 1 to the exponent (making it 9) and divide by the new exponent.
- For the term [tex]\( 7x^6 \)[/tex], the antiderivative is:
[tex]\[
\frac{7}{7}x^7 = x^7
\][/tex]
Again, we add 1 to the exponent (making it 7) and divide by the new exponent.
- For the term [tex]\(-8x^2\)[/tex], the antiderivative is:
[tex]\[
\frac{-8}{3}x^3 = -\frac{8}{3}x^3
\][/tex]
Once more, add 1 to the exponent (making it 3) and divide by the new exponent.
- For the constant term [tex]\(-7\)[/tex], the antiderivative is:
[tex]\[
-7x
\][/tex]
Since the antiderivative of a constant [tex]\( c \)[/tex] is [tex]\( cx \)[/tex].
2. Combine all the antiderivatives together:
Now that we have integrated each term, we combine them to find the complete antiderivative of the function. Adding all these results together gives us:
[tex]\[
\frac{2}{3}x^9 + x^7 - \frac{8}{3}x^3 - 7x
\][/tex]
3. Don't forget the constant of integration [tex]\( C \)[/tex]:
Since we are finding an indefinite integral, we need to include the constant of integration [tex]\( C \)[/tex]. Therefore, the final answer is:
[tex]\[
\frac{2}{3}x^9 + x^7 - \frac{8}{3}x^3 - 7x + C
\][/tex]
So, the antiderivative of [tex]\( f(x) = 6x^8 + 7x^6 - 8x^2 - 7 \)[/tex] is [tex]\(\frac{2}{3}x^9 + x^7 - \frac{8}{3}x^3 - 7x + C\)[/tex].
1. Integrate each term separately:
- For the term [tex]\( 6x^8 \)[/tex], the antiderivative is:
[tex]\[
\frac{6}{9}x^9 = \frac{2}{3}x^9
\][/tex]
Here, we add 1 to the exponent (making it 9) and divide by the new exponent.
- For the term [tex]\( 7x^6 \)[/tex], the antiderivative is:
[tex]\[
\frac{7}{7}x^7 = x^7
\][/tex]
Again, we add 1 to the exponent (making it 7) and divide by the new exponent.
- For the term [tex]\(-8x^2\)[/tex], the antiderivative is:
[tex]\[
\frac{-8}{3}x^3 = -\frac{8}{3}x^3
\][/tex]
Once more, add 1 to the exponent (making it 3) and divide by the new exponent.
- For the constant term [tex]\(-7\)[/tex], the antiderivative is:
[tex]\[
-7x
\][/tex]
Since the antiderivative of a constant [tex]\( c \)[/tex] is [tex]\( cx \)[/tex].
2. Combine all the antiderivatives together:
Now that we have integrated each term, we combine them to find the complete antiderivative of the function. Adding all these results together gives us:
[tex]\[
\frac{2}{3}x^9 + x^7 - \frac{8}{3}x^3 - 7x
\][/tex]
3. Don't forget the constant of integration [tex]\( C \)[/tex]:
Since we are finding an indefinite integral, we need to include the constant of integration [tex]\( C \)[/tex]. Therefore, the final answer is:
[tex]\[
\frac{2}{3}x^9 + x^7 - \frac{8}{3}x^3 - 7x + C
\][/tex]
So, the antiderivative of [tex]\( f(x) = 6x^8 + 7x^6 - 8x^2 - 7 \)[/tex] is [tex]\(\frac{2}{3}x^9 + x^7 - \frac{8}{3}x^3 - 7x + C\)[/tex].
