Find all possible functions with the given derivative.
-$y ^ { \prime } = x ^ { 3 } - 8 x$
A) $\frac { x ^ { 4 } } { 4 } - \frac { 8 x ^ { 2 } } { 2 } + C$
B) $\frac { x ^ { 4 } } { 4 } + 8 x ^ { 2 } + C$
C) $\frac { x ^ { 3 } } { 3 } - \frac { 8 x ^ { 2 } } { 2 } + C$
D) $3 x ^ { 2 } - 8 + C$

Question

Quizplus · Accepted Answer

To find the antiderivative of $y' = x^3 - 8x$, integrate each term separately. The antiderivative of $x^3$ is $\frac{x^4}{4}$ and the antiderivative of $-8x$ is $-\frac{8x^2}{2}$. Adding a constant $C$ gives the general solution $\frac{x^4}{4} - 4x^2 + C$, which matches option A.

Find All Possible Functions with the Given Derivative y′=x3−8xy ^ { \prime } = x ^ { 3 } - 8 xy′=x3−8x

Find All Possible Functions with the Given Derivative $y ^ { \prime } = x ^ { 3 } - 8 x$