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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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marked as duplicate by Simply Beautiful Art calculus Aug 16 '18 at 13:40

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  • $\begingroup$ hint: use Gamma function $\endgroup$ – pointguard0 Aug 16 '18 at 11:26
  • $\begingroup$ I tried that, could you be more specific? $\endgroup$ – Jonathan Kiersch Aug 16 '18 at 11:28
  • $\begingroup$ Hint 2. For $0 < x < 1$ and $\operatorname{Re}(z) \geq 0$ with $z \neq 0$, we have $$ \int_{0}^{\infty} t^{x-1} e^{-zt} \, dt = \frac{\Gamma(x)}{z^x}. $$ $\endgroup$ – Sangchul Lee Aug 16 '18 at 11:39
  • $\begingroup$ I don't know how to integrate complex functions $\endgroup$ – Jonathan Kiersch Aug 16 '18 at 11:47
  • $\begingroup$ See also: Generalized Fresnel integral. $\endgroup$ – Simply Beautiful Art Aug 16 '18 at 13:37