For each of Q , S , V in the following sentences, choose the appropriate expression from among (0) $\sim$ (7) at the bottom of this page. For the other $\square$, enter the correct number.

Suppose we have a differentiable function $f ( x )$ which satisfies the equation

$$\int _ { 0 } ^ { x } f ( t ) d t = \left( 1 + e ^ { - x } \right) f ( x ) + 2 x - 4 \log 2 \tag{1}$$

We are to find $f ( x )$ and the value of $\lim _ { x \rightarrow \infty } f ( x )$.

When we differentiate each side of (1) with respect to $x$ and transform the equation, we have

$$\left( 1 + e ^ { - x } \right) ( \mathbf { Q } ) = \mathbf { R } . \tag{2}$$

Next we set $f ( x ) = e ^ { x } g ( x )$, and using (2), we obtain

$$g ^ { \prime } ( x ) = \frac { \mathbf { S } } { 1 + e ^ { - x } }$$

and hence

$$g ( x ) = \mathbf { T } \log \left( 1 + e ^ { - x } \right) + C ,$$

where $C$ is an integral constant.\\
Furthermore, since $g ( 0 ) = f ( 0 )$, we see that $C = \mathbf { U }$. Thus we obtain $g ( x )$ and from that,

$$f ( x ) = \mathbf { V } \log \left( 1 + e ^ { - x } \right) .$$

Finally, we set $e ^ { - x } = t$ and obtain

$$f ( x ) = \mathbf { W } \log ( 1 + t ) ^ { \frac { 1 } { t } }$$

and hence

$$\lim _ { x \rightarrow \infty } f ( x ) = \lim _ { t \rightarrow \mathbf { X } } \mathbf { W } \log ( 1 + t ) ^ { \frac { 1 } { t } } = \mathbf { Y }$$

Choices:\\
(0) $f ^ { \prime } ( x ) - f ( x )$\\
(1) $f ( x ) - f ^ { \prime } ( x )$\\
(2) $f ^ { \prime } ( x ) - 2 f ( x )$\\
(3) $f ( x ) - 2 f ^ { \prime } ( x )$\\
(4) $2 e ^ { x }$\\
(5) $- 2 e ^ { x }$\\
(6) $2 e ^ { - x }$\\
(7) $- 2 e ^ { - x }$