Question

find the antiderivative for each function when c equals 0. check your answers by differentiation.
(a) ( g(x)=\frac{1}{x^{4}} )
(b) ( h(x)=\frac{5}{x^{4}} )
(c) ( k(x)=3-\frac{5}{x^{4}} )
(a) ( g(x)= )

Explanation:

Step1: Rewrite the function

Rewrite \(g(x)=\frac{1}{x^{4}}\) as \(g(x)=x^{- 4}\).

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 \(n=-4\), we have \(\int x^{-4}dx=\frac{x^{-4 + 1}}{-4+1}+C=\frac{x^{-3}}{-3}+C=-\frac{1}{3x^{3}}+C\).
Since \(C = 0\), \(G(x)=-\frac{1}{3x^{3}}\).

Step3: Check by differentiation

Differentiate \(G(x)=-\frac{1}{3x^{3}}=-\frac{1}{3}x^{-3}\) using the power rule \((x^{n})^\prime=nx^{n - 1}\).
\(G^\prime(x)=-\frac{1}{3}\times(-3)x^{-3-1}=x^{-4}=\frac{1}{x^{4}}=g(x)\).

Answer:

\(-\frac{1}{3x^{3}}\)