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 ( square ).
(use parentheses to clearly denote the argument of each function.)

Explanation:

Step1: Use the constant - multiple rule of integration

The constant - multiple rule states that if \(y = k\cdot f(x)\), then \(\int k\cdot f(x)dx=k\int f(x)dx\), where \(k\) is a constant. For \(g(x)=\frac{8}{x}\), we can write \(\int\frac{8}{x}dx = 8\int\frac{1}{x}dx\).

Step2: Use the antiderivative formula for \(\frac{1}{x}\)

We know that the antiderivative of \(\frac{1}{x}\) is \(\ln|x|\) (i.e., \(\int\frac{1}{x}dx=\ln|x|+C\)). When \(C = 0\), substituting the antiderivative of \(\frac{1}{x}\) into the expression from Step 1, we get \(8\int\frac{1}{x}dx=8\ln|x|\).

Answer:

\(8\ln|x|\)