Question

find the antiderivative for each function when c equals 0. check your answers by differentiation
(a) ( g(x)=\frac{1}{x^{7}} )
(b) ( h(x)=\frac{5}{x^{7}} )
(c) ( k(x)=4-\frac{5}{x^{7}} )
(a) ( g(x)= )

Explanation:

Step1: Rewrite the function

Rewrite \(g(x)=\frac{1}{x^{7}}\) as \(g(x)=x^{-7}\).

Step2: 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 \(n=-7\), we have \(\int x^{-7}dx=\frac{x^{-7 + 1}}{-7+1}+C\).

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Since \(C = 0\), \(G(x)=-\frac{1}{6x^{6}}\).

Step3: Check by differentiation

Differentiate \(G(x)=-\frac{1}{6x^{6}}=-\frac{1}{6}x^{-6}\) using the power - rule \((x^{n})^\prime=nx^{n - 1}\).
\(G^\prime(x)=-\frac{1}{6}\times(-6)x^{-6-1}=x^{-7}=\frac{1}{x^{7}}=g(x)\)

Answer:

\(-\frac{1}{6x^{6}}\)