6. 0 / 10 points 4/100 submissions used\nfind f. (use c for the constant of the first antiderivative and d…

6. 0 / 10 points 4/100 submissions used\nfind f. (use c for the constant of the first antiderivative and d for the constant of the second antiderivative.)\n$f(x)=24x^{3}-18x^{2}+8x$\n$f(x)=\\frac{6}{5}x^{5}-\\frac{3}{2}x^{4}+(\\frac{4}{3})^{3}+cx+d$\nresources

6. 0 / 10 points 4/100 submissions used\nfind f. (use c for the constant of the first antiderivative and d for the constant of the second antiderivative.)\n$f(x)=24x^{3}-18x^{2}+8x$\n$f(x)=\\frac{6}{5}x^{5}-\\frac{3}{2}x^{4}+(\\frac{4}{3})^{3}+cx+d$\nresources

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\neq - 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.