Sure, let's find the general antiderivative of the function [tex]\( f(x) = -5x^2 + \frac{5}{x} + \frac{6}{x^4} - 6\sqrt{x} \)[/tex].
To find the antiderivative [tex]\( \int f(x) \, dx \)[/tex], we will integrate each term of the function separately.
1. Integrate [tex]\(-5x^2 \)[/tex]:
[tex]\[ \int -5x^2 \, dx \][/tex]
Using the power rule for integration, [tex]\(\int x^n \, dx = \frac{x^{n+1}}{n+1}\)[/tex], we get:
[tex]\[ \int -5x^2 \, dx = -5 \cdot \frac{x^{2+1}}{2+1} = -5 \cdot \frac{x^3}{3} = -\frac{5}{3}x^3 \][/tex]
2. Integrate [tex]\( \frac{5}{x} \)[/tex]:
[tex]\[ \int \frac{5}{x} \, dx \][/tex]
This is a standard integral, [tex]\(\int \frac{1}{x} \, dx = \ln|x|\)[/tex], so:
[tex]\[ \int \frac{5}{x} \, dx = 5 \ln|x| \][/tex]
3. Integrate [tex]\( \frac{6}{x^4} \)[/tex]:
[tex]\[ \int \frac{6}{x^4} \, dx \][/tex]
Rewrite [tex]\(\frac{6}{x^4}\)[/tex] as [tex]\(6x^{-4}\)[/tex]. Using the power rule for integration, we get:
[tex]\[ \int 6x^{-4} \, dx = 6 \cdot \frac{x^{-4+1}}{-4+1} = 6 \cdot \frac{x^{-3}}{-3} = -2x^{-3} = -\frac{2}{x^3} \][/tex]
4. Integrate [tex]\(-6\sqrt{x} \)[/tex]:
[tex]\[ \int -6\sqrt{x} \, dx \][/tex]
Rewrite [tex]\(\sqrt{x}\)[/tex] as [tex]\(x^{1/2}\)[/tex]. Using the power rule for integration, we get:
[tex]\[ \int -6x^{1/2} \, dx = -6 \cdot \frac{x^{1/2+1}}{1/2+1} = -6 \cdot \frac{x^{3/2}}{3/2} = -6 \cdot \frac{2x^{3/2}}{3} = -4x^{3/2} \][/tex]
Now, combine all these results to find the general antiderivative of the function:
[tex]\[ \int f(x) \, dx = -\frac{5}{3}x^3 + 5\ln|x| - \frac{2}{x^3} - 4x^{3/2} + C \][/tex]
Therefore, the general antiderivative of the function [tex]\( f(x) = -5x^2 + \frac{5}{x} + \frac{6}{x^4} - 6\sqrt{x} \)[/tex] is:
[tex]\[ \boxed{ -\frac{5}{3}x^3 + 5\ln|x| - \frac{2}{x^3} - 4x^{3/2} + C } \][/tex]
