Evaluate integral of Sine function

[muaythai2006]

Hello guyz, I am new at this page. I need your help. I can't able to evaluate this integral. intergral of( sin(pi*t/(2T)) e^ -j2*pi*f*t)dt . The lower limit is 0 and the upper limit is 2T ...This is acctually the Fourier transform of sin(pi*t/(2T)) where 0<t<2T ...I could do the Fourier transform of this but only if the limit of t is -infinity to infinity...Thanks for your help...

 

[vsage]

Are you familiar with Euler's identity? Consider rewriting $$sin(\frac{ \pi t}{2T} )$$ as $$\frac{1}{2j} (e^{\frac{j \pi t}{2T}} - e^{-\frac{j \pi t}{2T}})$$

(Edit: Fixing the equation, LaTeX)