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find f. (use c for the constant of the first antiderivative and d for the constant of the second antiderivative.)
$f(x)=24x^{3}-18x^{2}+8x$
$f(x)=\frac{6}{5}x^{5}-\frac{3}{2}x^{4}+(\frac{4}{3})^{3}+cx+d$
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Accepted Answer

# Answer:
$f(x)=6x^{4}-6x^{3}+4x^{2}+Cx + D$
# Explanation:
## Step1: Integrate $f''(x)$ to find $f'(x)$
Use the power rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n
eq - 1$).
$$
\begin{align*}
f'(x)&=\int(24x^{3}-18x^{2}+8x)dx\
&=24\int x^{3}dx-18\int x^{2}dx + 8\int xdx\
&=24\times\frac{x^{4}}{4}-18\times\frac{x^{3}}{3}+8\times\frac{x^{2}}{2}+C\
&=6x^{4}-6x^{3}+4x^{2}+C
\end{align*}
$$
## Step2: Integrate $f'(x)$ to find $f(x)$
$$
\begin{align*}
f(x)&=\int(6x^{4}-6x^{3}+4x^{2}+C)dx\
&=6\int x^{4}dx-6\int x^{3}dx+4\int x^{2}dx+\int Cdx\
&=6\times\frac{x^{5}}{5}-6\times\frac{x^{4}}{4}+4\times\frac{x^{3}}{3}+Cx+D\
&=\frac{6}{5}x^{5}-\frac{3}{2}x^{4}+\frac{4}{3}x^{3}+Cx + D
\end{align*}
$$
The error in the original answer was in the integration of $8x$ in the first - step (when finding $f'(x)$) and then carrying the wrong power (the $x$ in the $(\frac{4}{3})^{3}$ term, which should be $x^{3}$). The correct antiderivative is obtained by properly applying the power - rule for integration at each step.

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QUESTION IMAGE

Question

Explanation:

Step1: Integrate \(f''(x)\) to find \(f'(x)\)

Step2: Integrate \(f'(x)\) to find \(f(x)\)

Answer: