Let $f ( x ) = \left\{ \begin{array} { c c c } 0 & \text { if } & x 3 \end{array} \right\}$ . The Fourier integral representation of f is A) $f ( x ) = \int _ { 0 } ^ { \infty } \{ [ \sin ( \alpha x ) + \sin ( ( 3 - x ) \alpha ) ] / \alpha \} d \alpha / \pi$ B) $f ( x ) = 2 \int _ { 0 } ^ { \infty } \{ [ \sin ( \alpha x ) + \sin ( ( 3 - x ) \alpha ) ] / \alpha \} d \alpha$ C) $f ( x ) = 2 \int _ { 0 } ^ { \infty } \{ [ \sin ( \alpha x ) + \sin ( ( 3 - x ) \alpha ) ] / \alpha \} d \alpha / \pi$ D) $$f ( x ) = \int _ { 0 } ^ { \infty } \{ [ \sin ( a x ) + \sin ( ( 3 - x ) \alpha ) ] / \alpha \} d \alpha$$ E) none of the above

Question

Let $f ( x ) = \left\{ \begin{array} { c c c } 
0 & \text { if } & x < 0 \
2 & \text { if } & 0 \leq x \leq 3 \
0 & \text { if } & x > 3
\end{array} \right\}$ . The Fourier integral representation of f is
A) $f ( x ) = \int _ { 0 } ^ { \infty } \{ [ \sin ( \alpha x ) + \sin ( ( 3 - x ) \alpha ) ] / \alpha \} d \alpha / \pi$
B) $f ( x ) = 2 \int _ { 0 } ^ { \infty } \{ [ \sin ( \alpha x ) + \sin ( ( 3 - x ) \alpha ) ] / \alpha \} d \alpha$
C) $f ( x ) = 2 \int _ { 0 } ^ { \infty } \{ [ \sin ( \alpha x ) + \sin ( ( 3 - x ) \alpha ) ] / \alpha \} d \alpha / \pi$
D) $$f ( x ) = \int _ { 0 } ^ { \infty } \{ [ \sin ( a x ) + \sin ( ( 3 - x ) \alpha ) ] / \alpha \} d \alpha$$
E) none of the above

Quizplus · Accepted Answer

The Answer is C

Let the Fourier Integral Representation of F Is A) f(x)=∫0∞{[sin⁡(αx)+sin⁡((3−x)α)]/α}dα/πf ( x ) = \int _ { 0 } ^ { \infty } \{ [ \sin ( \alpha x ) + \sin ( ( 3 - x ) \alpha ) ] / \alpha \} d \alpha / \pif(x)=∫0∞​{[sin(αx)+sin((3−x)α)]/α}dα/π

Let the Fourier Integral Representation of F Is
A) $f ( x ) = \int _ { 0 } ^ { \infty } \{ [ \sin ( \alpha x ) + \sin ( ( 3 - x ) \alpha ) ] / \alpha \} d \alpha / \pi$