[Calculus] For two functions $f ( x )$ and $g ( x )$ that have second derivatives on the set of all real numbers, consider the definite integral $$\int _ { 0 } ^ { 1 } \left\{ f ^ { \prime } ( x ) g ( 1 - x ) - g ^ { \prime } ( x ) f ( 1 - x ) \right\} d x$$ Let the value of this integral be $k$. Which of the following statements in are correct? [4 points] ㄱ. $\int _ { 0 } ^ { 1 } \left\{ f ( x ) g ^ { \prime } ( 1 - x ) - g ( x ) f ^ { \prime } ( 1 - x ) \right\} d x = - k$ ㄴ. If $f ( 0 ) = f ( 1 )$ and $g ( 0 ) = g ( 1 )$, then $k = 0$. ㄷ. If $f ( x ) = \ln \left( 1 + x ^ { 4 } \right)$ and $g ( x ) = \sin \pi x$, then $k = 0$. (1) ㄴ (2) ㄷ (3) ㄱ, ㄴ (4) ㄱ, ㄷ (5) ㄱ, ㄴ, ㄷ
[Calculus] For two functions $f ( x )$ and $g ( x )$ that have second derivatives on the set of all real numbers, consider the definite integral
$$\int _ { 0 } ^ { 1 } \left\{ f ^ { \prime } ( x ) g ( 1 - x ) - g ^ { \prime } ( x ) f ( 1 - x ) \right\} d x$$
Let the value of this integral be $k$. Which of the following statements in <Remarks> are correct? [4 points]
\textbf{<Remarks>}\\
ㄱ. $\int _ { 0 } ^ { 1 } \left\{ f ( x ) g ^ { \prime } ( 1 - x ) - g ( x ) f ^ { \prime } ( 1 - x ) \right\} d x = - k$\\
ㄴ. If $f ( 0 ) = f ( 1 )$ and $g ( 0 ) = g ( 1 )$, then $k = 0$.\\
ㄷ. If $f ( x ) = \ln \left( 1 + x ^ { 4 } \right)$ and $g ( x ) = \sin \pi x$, then $k = 0$.\\
(1) ㄴ\\
(2) ㄷ\\
(3) ㄱ, ㄴ\\
(4) ㄱ, ㄷ\\
(5) ㄱ, ㄴ, ㄷ