# 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

## marked as duplicate by Simply Beautiful Art calculus StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Aug 16 '18 at 13:40

• hint: use Gamma function – pointguard0 Aug 16 '18 at 11:26
• I tried that, could you be more specific? – Jonathan Kiersch Aug 16 '18 at 11:28
• 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
• I don't know how to integrate complex functions – Jonathan Kiersch Aug 16 '18 at 11:47
• See also: Generalized Fresnel integral. – Simply Beautiful Art Aug 16 '18 at 13:37