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}} )
(a) ( h(x)=\frac{1}{2} x^{\frac{8}{9}} )
(b) ( g(x)= )

Explanation:

Step1: Apply 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 \(g(x)=\frac{1}{9}x^{-\frac{4}{9}}\), we have \(n=-\frac{4}{9}\).

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Step2: Calculate the integral

Using the power rule \(\int x^{n}dx=\frac{x^{n + 1}}{n + 1}\) (\(C = 0\)), with \(n=-\frac{4}{9}\), we get:

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Step3: Check by differentiation

Differentiate \(G(x)=\frac{1}{5}x^{\frac{5}{9}}\) using the power rule \(\frac{d}{dx}(x^{n})=nx^{n - 1}\).

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Answer:

\(G(x)=\frac{1}{5}x^{\frac{5}{9}}\)