Question

find the antiderivative for each function when c equals 0. check your answers by differentiatio
(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)=-\frac{1}{6 x^{6}} )
(b) ( h(x)= )

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 \(y = \frac{5}{x^{7}}=5x^{-7}\), we can use the power - rule.

Step2: Apply the power - rule

When \(n=-7\), \(\int5x^{-7}dx=5\int x^{-7}dx\).
Using the formula \(\int x^{n}dx=\frac{x^{n + 1}}{n + 1}+C\) (\(C = 0\) in our case), we have \(5\times\frac{x^{-7+1}}{-7 + 1}\).

$$
LATEXBLOCK0
$$

Step3: Check by differentiation

Differentiate \(H(x)=-\frac{5}{6x^{6}}=-\frac{5}{6}x^{-6}\) using the power - rule for differentiation \((x^{n})^\prime=nx^{n-1}\).
\(H^\prime(x)=-\frac{5}{6}\times(-6)x^{-6 - 1}\).

$$
LATEXBLOCK1
$$

Answer:

\(-\frac{5}{6x^{6}}\)