Find a function f(x) with the following property: f'(x) = 2 $\mathrm { e } ^ { 2 x }$ - $\frac {4} { \mathrm { x }}$ + $\frac { 3 } { x ^ { 2 } }$ , f(1) = -2.
A) f(x) = $\mathrm { e } ^ { 2 x }$ - 4 ln|x| - $\frac { 3 } { x }$ - ( $e ^ { 2 }$ - 1)
B) f(x) = $\mathrm { e } ^ { 2 x }$ - 4 ln|x| - $\frac { 6 x } { x ^ { 3 } }$ + 3 + 4e
C) f(x) = $\frac { 2 } { 3 }$ $e ^ { 3 x }$ - 4x - $\frac { 1 } { x }$ + $\frac { 5 } { 3 }$
D) f(x) = $\frac { 2 } { 3 }$ $e ^ { 3 x }$ - 4 ln|x| - $\frac { 1 } { x }$ - $\frac { 5 } { 3 }$

Question

Quizplus · Accepted Answer

To find the antiderivative of $f'(x) = 2e^{2x} - \frac{4}{x} + \frac{3}{x^2}$, we integrate each term separately. The antiderivative of $2e^{2x}$ is $e^{2x}$, of $-\frac{4}{x}$ is $-4\ln|x|$, and of $\frac{3}{x^2}$ is $-\frac{3}{x}$. Adding a constant $C$ and using the condition $f(1) = -2$ to solve for $C$ gives us choice A.

Find a Function F(x) with the Following Property: F'(x) = e2x\mathrm { e } ^ { 2 x }e2x

Find a Function F(x) with the Following Property: F'(x) = $\mathrm { e } ^ { 2 x }$