To find the antiderivative [tex]\( F(x) \)[/tex] of the function [tex]\( f(x) = \frac{2}{x^2} + 8 \cdot x^6 \)[/tex], we follow these steps:
1. Break down the function into simpler parts:
[tex]\[
f(x) = \frac{2}{x^2} + 8 \cdot x^6
\][/tex]
2. Rewrite the function in terms of exponents:
[tex]\[
f(x) = 2 \cdot x^{-2} + 8 \cdot x^6
\][/tex]
3. Integrate each term separately:
- For [tex]\( 2 \cdot x^{-2} \)[/tex]:
The antiderivative of [tex]\( x^{-n} \)[/tex] where [tex]\( n \neq 1 \)[/tex] is [tex]\( \frac{x^{-n+1}}{-n+1} \)[/tex].
Here, [tex]\( n = 2 \)[/tex], so:
[tex]\[
\int 2 \cdot x^{-2} \, dx = 2 \cdot \frac{x^{-2+1}}{-2+1} = 2 \cdot \frac{x^{-1}}{-1} = -2 \cdot x^{-1} = -\frac{2}{x}
\][/tex]
- For [tex]\( 8 \cdot x^6 \)[/tex]:
The antiderivative of [tex]\( x^n \)[/tex] is [tex]\( \frac{x^{n+1}}{n+1} \)[/tex].
Here, [tex]\( n = 6 \)[/tex], so:
[tex]\[
\int 8 \cdot x^6 \, dx = 8 \cdot \frac{x^{6+1}}{6+1} = 8 \cdot \frac{x^7}{7} = \frac{8x^7}{7}
\][/tex]
4. Combine the results:
[tex]\[
F(x) = \left( -\frac{2}{x} \right) + \left( \frac{8x^7}{7} \right)
\][/tex]
5. Add the constant of integration [tex]\( C \)[/tex]:
[tex]\[
F(x) = \frac{8 \cdot x^7}{7} - \frac{2}{x} + C
\][/tex]
So, the antiderivative of [tex]\( f(x)=\frac{2}{x^2}+8 \cdot x^6 \)[/tex] is:
[tex]\[
F(x) = \boxed{\frac{8 \cdot x^7}{7} - \frac{2}{x}} + C
\][/tex]
