Nice integral computed with the assistance of Mellin transform of sine and cosine

The goal is to compute the following Integral


I=\int_{0}^{\infty} \frac{\left(\sin ( x)- \cos(x) \right) \ln x}{\sqrt{x}} d x


Lets start by considering the pair of Mellin Transforms proved here:

\int_{0}^{\infty} \cos ( x) x^{a-1} d x &=\Gamma(a) \cos \left(\frac{a \pi}{2}\right)

\int_{0}^{\infty} \sin ( x) x^{a-1} d x &=\Gamma(a) \sin \left(\frac{a \pi}{2}\right)

\int_{0}^{\infty} \cos ( x) x^{a}x^{-1} d x &=\Gamma(a) \cos \left(\frac{a \pi}{2}\right)

\int_{0}^{\infty} \sin ( x) x^{a}x^{-1} d x &=\Gamma(a) \sin \left(\frac{a \pi}{2}\right)

\int_{0}^{\infty} \cos ( x) e^{a \ln x} x^{-1} d x &=\Gamma(a) \cos \left(\frac{a \pi}{2}\right)

\int_{0}^{\infty} \sin ( x) e^{a \ln x}x^{-1} d x &=\Gamma(a) \sin \left(\frac{a \pi}{2}\right)

\frac{d}{da}\int_{0}^{\infty} \cos ( x) e^{a \ln x} x^{-1} d x &=\Gamma^{\prime}(a) \cos \left(\frac{a \pi}{2}\right)- \frac{\pi}{2}\Gamma(a) \sin \left(\frac{a \pi}{2}\right)

\frac{d}{da}\int_{0}^{\infty} \sin ( x) e^{a \ln x}x^{-1} d x &=\Gamma^{\prime}(a) \sin \left(\frac{a \pi}{2}\right)+\frac{\pi}{2}\Gamma(a) \cos \left(\frac{a \pi}{2}\right)

\int_{0}^{\infty} \cos ( x) x^{a-1}\ln x d x &=\Gamma^{\prime}(a) \cos \left(\frac{a \pi}{2}\right)- \frac{\pi}{2}\Gamma(a) \sin \left(\frac{a \pi}{2}\right)(1)

\int_{0}^{\infty} \sin ( x) x^{a-1}\ln x d x &=\Gamma^{\prime}(a) \sin \left(\frac{a \pi}{2}\right)+\frac{\pi}{2}\Gamma(a) \cos \left(\frac{a \pi}{2}\right)(2)

set   a=\frac{1}{2}   in (1) and (2)


\int_{0}^{\infty} \frac{\cos ( x) \ln x}{\sqrt{x}} d x &=\Gamma^{\prime}\left(\frac{1}{2}\right) \cos \left(\frac{ \pi}{4}\right)- \frac{\pi}{2}\Gamma\left(\frac{1}{2}\right) \sin \left(\frac{ \pi}{4}\right)(3)

\int_{0}^{\infty} \frac{\sin ( x) \ln x}{\sqrt{x}} d x &=\Gamma^{\prime}\left(\frac{1}{2}\right) \sin \left(\frac{ \pi}{4}\right)+\frac{\pi}{2}\Gamma\left(\frac{1}{2}\right) \cos \left(\frac{ \pi}{4}\right)(4)


Now subtract (3) from(4)


\int_{0}^{\infty} \frac{\left(\sin ( x)- \cos(x) \right) \ln x}{\sqrt{x}} d x &=\Gamma^{\prime}\left(\frac{1}{2}\right) \underbrace{\left(\sin \left(\frac{ \pi}{4}\right)- \cos \left(\frac{ \pi}{4}\right)\right)}_{=0}+\frac{\pi}{2}\Gamma\left(\frac{1}{2}\right) \left(\cos \left(\frac{ \pi}{4}\right)+\sin \left(\frac{ \pi}{4}\right)\right)


\int_{0}^{\infty} \frac{\left(\sin ( x)- \cos(x) \right) \ln x}{\sqrt{x}} d x &=\frac{\pi}{2}\sqrt{\pi} \left(2\frac{\sqrt{2}}{2}\right)


\boxed{\int_{0}^{\infty} \frac{\left(\sin ( x)- \cos(x) \right) \ln x}{\sqrt{x}} d x &=\frac{\pi^{3/2}}{\sqrt{2}}}

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