To find the general antiderivative, also known as the indefinite integral, of the function [tex]\( f(x) = 2x^6 - \frac{5}{x} - \frac{6}{x^5} + 5\sqrt{x} \)[/tex], we will integrate each term individually with respect to [tex]\( x \)[/tex].
We begin by separating the function into its individual terms:
[tex]\[ f(x) = 2x^6 - \frac{5}{x} - \frac{6}{x^5} + 5\sqrt{x} \][/tex]
Now, we integrate each term one by one.
1. First Term: [tex]\( 2x^6 \)[/tex]
[tex]\[ \int 2x^6 \, dx \][/tex]
The integral of [tex]\( x^n \)[/tex] is [tex]\( \frac{x^{n+1}}{n+1} \)[/tex]:
[tex]\[ \int 2x^6 \, dx = 2 \cdot \frac{x^{6+1}}{6+1} = 2 \cdot \frac{x^7}{7} = \frac{2x^7}{7} \][/tex]
2. Second Term: [tex]\( -\frac{5}{x} \)[/tex]
[tex]\[ \int -\frac{5}{x} \, dx \][/tex]
The integral of [tex]\( \frac{1}{x} \)[/tex] is [tex]\( \ln|x| \)[/tex]:
[tex]\[ \int -\frac{5}{x} \, dx = -5 \int \frac{1}{x} \, dx = -5 \ln|x| \][/tex]
For ease, we typically drop the absolute value when assuming [tex]\( x \)[/tex] is positive:
[tex]\[ -5 \ln|x| = -5 \log(x) \][/tex]
3. Third Term: [tex]\( -\frac{6}{x^5} \)[/tex]
[tex]\[ \int -\frac{6}{x^5} \, dx \][/tex]
Rewrite [tex]\( \frac{1}{x^5} \)[/tex] as [tex]\( x^{-5} \)[/tex]:
[tex]\[ \int -6x^{-5} \, dx \][/tex]
Integrate using the rule for powers of [tex]\( x \)[/tex]:
[tex]\[ \int -6x^{-5} \, dx = -6 \cdot \frac{x^{-5+1}}{-5+1} = -6 \cdot \frac{x^{-4}}{-4} = \frac{-6x^{-4}}{-4} = \frac{6}{4}x^{-4} = \frac{3}{2}x^{-4} \][/tex]
4. Fourth Term: [tex]\( 5\sqrt{x} \)[/tex]
[tex]\[ \int 5\sqrt{x} \, dx \][/tex]
Rewrite [tex]\( \sqrt{x} \)[/tex] as [tex]\( x^{1/2} \)[/tex]:
[tex]\[ \int 5x^{1/2} \, dx \][/tex]
Integrate using the rule for powers of [tex]\( x \)[/tex]:
[tex]\[ \int 5x^{1/2} \, dx = 5 \cdot \frac{x^{1/2 + 1}}{1/2 + 1} = 5 \cdot \frac{x^{3/2}}{3/2} = 5 \cdot \frac{2x^{3/2}}{3} = \frac{10}{3}x^{3/2} \][/tex]
Now, combine all the results and add the constant of integration [tex]\( C \)[/tex]:
[tex]\[ \int f(x) \, dx = \frac{2x^7}{7} - 5 \log(x) + \frac{3}{2}x^{-4} + \frac{10x^{3/2}}{3} + C \][/tex]
Simplify the expression:
[tex]\[ \int f(x) \, dx = \frac{2x^7}{7} - 5 \log(x) + \frac{3}{2x^4} + \frac{10x^{3/2}}{3} + C \][/tex]
Thus, the general antiderivative of [tex]\( f(x) \)[/tex] is:
[tex]\[ \boxed{ \frac{2x^7}{7} - 5 \log(x) + \frac{3}{2x^4} + \frac{10x^{3/2}}{3} + C } \][/tex]
