Find all possible functions with the given derivative.
-$\mathbf { r } ^ { \prime } = 4 + \frac { 1 } { \theta ^ { 4 } }$
A) $r = 4 \theta + \theta ^ { 4 }$
B) $r = 4 \theta - \theta ^ { 5 }$
C) $r = 4 \theta - \frac { 1 } { 3 \theta ^ { 3 } }$
D) $r = 4 \theta + \frac { 1 } { 5 \theta ^ { 5 } }$

Question

Quizplus · Accepted Answer

To find the antiderivative of $4 + \frac{1}{\theta^4}$, integrate each term separately. The antiderivative of $4$ with respect to $\theta$ is $4\theta$, and the antiderivative of $\frac{1}{\theta^4}$ is $-\frac{1}{3\theta^3}$ (using the power rule for integration, $\int \theta^n d\theta = \frac{\theta^{n+1}}{n+1} + C$ for $n \neq -1$). Therefore, the correct function is $r = 4\theta - \frac{1}{3\theta^3}$ plus a constant, which matches option C.

Find All Possible Functions with the Given Derivative r′=4+1θ4\mathbf { r } ^ { \prime } = 4 + \frac { 1 } { \theta ^ { 4 } }r′=4+θ41​

Find All Possible Functions with the Given Derivative $\mathbf { r } ^ { \prime } = 4 + \frac { 1 } { \theta ^ { 4 } }$