find the most general antiderivative of the function. (check your answer by differentiation. use c for the constant of the antiderivative.)
g(x) = 2x^{-2/3} - 2x^{5/3}
g(x) =

Question

Accepted Answer

# Explanation:
## Step1: Apply power rule for integration
The power rule for integration is $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$, where $n
eq - 1$.
For the first term $\int2x^{-\frac{2}{3}}dx$, using the power rule:
$$
\begin{align*}
\int2x^{-\frac{2}{3}}dx&=2\times\frac{x^{-\frac{2}{3}+1}}{-\frac{2}{3}+1}+C_1\
&=2\times\frac{x^{\frac{1}{3}}}{\frac{1}{3}}+C_1\
&=6x^{\frac{1}{3}}+C_1
\end{align*}
$$
For the second term $\int - 2x^{\frac{5}{3}}dx$, using the power rule:
$$
\begin{align*}
\int-2x^{\frac{5}{3}}dx&=-2\times\frac{x^{\frac{5}{3}+1}}{\frac{5}{3}+1}+C_2\
&=-2\times\frac{x^{\frac{8}{3}}}{\frac{8}{3}}+C_2\
&=-\frac{3}{4}x^{\frac{8}{3}}+C_2
\end{align*}
$$
## Step2: Combine the results
Combine the two integrated terms and the constants $C_1$ and $C_2$ into a single constant $C$:
$$G(x)=6x^{\frac{1}{3}}-\frac{3}{4}x^{\frac{8}{3}}+C$$

## Step3: Check by differentiation
Differentiate $G(x)$ using the power rule $(x^{n})^\prime=nx^{n - 1}$
$$
\begin{align*}
G^\prime(x)&=\frac{d}{dx}(6x^{\frac{1}{3}}-\frac{3}{4}x^{\frac{8}{3}}+C)\
&=6\times\frac{1}{3}x^{\frac{1}{3}-1}-\frac{3}{4}\times\frac{8}{3}x^{\frac{8}{3}-1}+0\
&=2x^{-\frac{2}{3}}-2x^{\frac{5}{3}}
\end{align*}
$$
Which is the original function $g(x)$

# Answer:
$G(x)=6x^{\frac{1}{3}}-\frac{3}{4}x^{\frac{8}{3}}+C$

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QUESTION IMAGE

Question

Explanation:

Step1: Apply power rule for integration

Step2: Combine the results

Step3: Check by differentiation

Answer: