The Fourier Cosine Integral Of $f ( x ) = e ^ { - x } , x \geq 0$

The Fourier cosine integral of $f ( x ) = e ^ { - x } , x \geq 0$ is

A) $f ( x ) = 2 \int _ { 0 } ^ { \infty } [ \alpha \cos ( \alpha x ) / ( 1 + \alpha ) ] d \alpha$
B) $f ( x ) = \int _ { 0 } ^ { \infty } \left[ \alpha \cos ( \alpha x ) / \left( 1 + \alpha ^ { 2 } \right) \right] d \alpha / \pi$
C) $f ( x ) = \int _ { 0 } ^ { \infty } \left[ \alpha \cos ( \alpha x ) / \left( 1 + \alpha ^ { 2 } \right) \right] d \alpha$
D) $f ( x ) = 2 \int _ { 0 } ^ { \infty } \left[ \cos ( \alpha x ) / \left( 1 + \alpha ^ { 2 } \right) \right] d \alpha / \pi$
E) $f ( x ) = 2 \int _ { 0 } ^ { \infty } [ \cos ( \alpha x ) / ( 1 + \alpha ) ] d \alpha / \pi$

Correct Answer:

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