The Fourier Integral Representation of a Function F Is Given $\int _ { - \infty } ^ { \infty } f ( x ) \cos ( \alpha x ) d x$

The Fourier integral representation of a function f is given by

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