Mathematical Yin-Yang

Calculus Level 4

If \(\frac { 1 }{ \sqrt { x } } =\frac { 2 }{ \sqrt { \pi } } \int _{ 0 }^{ \infty }{ e^{ -a^2x }d\alpha } , \) \(\alpha>0,\)

evaluate

\(\int _{ 0 }^{ \infty }{ \frac { cos{x}dx }{\sqrt{x} } } -\int _{ 0 }^{ \infty }{ \frac{\sin{x}dx}{\sqrt{x}} } \)

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