Question

find the antiderivative for each function when c equals 0. check your answers by differentiation.
(a) ( h(x)=\frac{4}{9} x^{-\frac{1}{9}} ) (b) ( g(x)=\frac{1}{9} x^{-\frac{4}{9}} ) (c) ( k(x)=-\frac{1}{9} x^{-\frac{10}{9}} )
( h(x)=square )

Explanation:

Step1: Recall the power - rule for integration

The power - rule for integration is \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)). For the function \(h(x)=\frac{4}{9}x^{-\frac{1}{9}}\), we have \(n=-\frac{1}{9}\).

Step2: Apply the power - rule

$$
LATEXBLOCK0
$$

Step3: Check by differentiation

Using the power - rule for differentiation \((x^{n})^\prime=nx^{n - 1}\), for \(y = \frac{1}{2}x^{\frac{8}{9}}\), we have \(y^\prime=\frac{1}{2}\times\frac{8}{9}x^{\frac{8}{9}-1}=\frac{4}{9}x^{-\frac{1}{9}}\)

Answer:

\(\frac{1}{2}x^{\frac{8}{9}}\)