Integration for special sine-function [duplicate]

I am having trouble integrating the following:

$\int_{0}^{\infty} \sin(t)\cdot t^{x-1} ~dt$ for $0 < x < 1$.

Does anybody know how that can be done. Ive tried to set it up with powerseries for $\sin(t)$, and by a substitution followed with integration by parts, non of that gave anything useful.

Does anybody know how to integrate this, using the gamma function?

Thanks

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• Hint 2. For $0 < x < 1$ and $\operatorname{Re}(z) \geq 0$ with $z \neq 0$, we have $$\int_{0}^{\infty} t^{x-1} e^{-zt} \, dt = \frac{\Gamma(x)}{z^x}.$$ – Sangchul Lee Aug 16 '18 at 11:39