Question

find the antiderivative for each function when c equals 0.
a. ( f(x)=\frac{1}{x} ) b. ( g(x)=\frac{8}{x} ) c. ( h(x)=5-\frac{8}{x} )
a. the antiderivative of ( \frac{1}{x} ) is ( ln |x| ).
(use parentheses to clearly denote the argument of each function.)
b. the antiderivative of ( \frac{8}{x} ) is ( 8 ln |x| ).
(use parentheses to clearly denote the argument of each function.)
c. the antiderivative of ( 5-\frac{8}{x} ) is
(use parentheses to clearly denote the argument of each function.)

Explanation:

Step1: Recall the antiderivative rules

The antiderivative of a constant \(k\) is \(kx\), and the antiderivative of \(\frac{a}{x}\) is \(a\ln|x|\).

Step2: Find the antiderivative of \(h(x)=5-\frac{8}{x}\)

For the term \(5\), using the rule for the antiderivative of a constant (\(k = 5\)), its antiderivative is \(5x\). For the term \(-\frac{8}{x}\), using the rule for the antiderivative of \(\frac{a}{x}\) (\(a=- 8\)), its antiderivative is \(-8\ln|x|\).
Combining these two results, the antiderivative of \(h(x)=5-\frac{8}{x}\) is \(5x-8\ln|x|\) (since \(C = 0\)).

Answer:

\(5x - 8\ln|x|\)