Mathematical Yin-Yang

Calculus Level 4

If \(\displaystyle \frac { 1 }{ \sqrt { x } } =\frac { 2 }{ \sqrt { \pi } } \int _{ 0 }^{ \infty }{ e^{ -a^2x }d\alpha }\) for \(\alpha>0\), evaluate \[\large \int _{ 0 }^{ \infty }{ \frac {\cos{x} \ dx }{\sqrt{x} } } -\int _{ 0 }^{ \infty }{ \frac{\sin{x} \ dx}{\sqrt{x}} } \]

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