Hints to calculate in a closed form $\int_0^1 \sin(2\pi k x)x^Ne^{-Nx}dx$, for integers $k,N\geq 1$ Let $k\geq 1$ an integer and $N\geq 1$, also an integer. I would like to know hints to get in a closed-form (I believe that it is possible find it) for $N\geq 1$ $$\int_0^1 \sin(2\pi k x)x^Ne^{-Nx}dx.$$
I know that I need define a definite integral $$I_{k,N}$$ integration by parts (what?, it isn't required prove the recurrence that one should deduce from the definition by mathematical induction). I know the first cases.

Question. How one calculate in a closed-form $$\int_0^1 \sin(2\pi k x)x^Ne^{-Nx} dx$$for integers $k,N\geq 1$?

 A: We have,  $$\sin (2 \pi k x) = \Im (e^{2 i \pi k x})$$ 
(where $\Im$ denotes the imaginary part)
so that,  
$$ I_{k,N} = \Im \left( \int_{0}^{1} x^N e^{-x (N-2i \pi k )} \ \mathrm{d}x \right)$$ 
Consider the Lower Incomplete Gamma Function,   
$$\gamma(a,b) = \int_{0}^{b} x^{a-1} e^{-x} \ \mathrm{d}x $$ 
Substitute $x \mapsto bx$ 
$$ \implies \gamma(a,b) = b^a \int_{0}^{1} x^{a-1} e^{-bx} \ \mathrm{d}x$$ 
$$ \implies \int_{0}^{1} x^{a-1} e^{-bx} \ \mathrm{d}x = \dfrac{\gamma(a,b)}{b^a} $$ 
$$ \therefore \ I_{k,N} = \Im \left(\dfrac{\gamma(N+1,N-2ik \pi x)}{(N-2ik \pi x)^{N+2}}\right) $$ 

Added : As pointed out in the comments, there is an elementary closed form for $N \in \mathbb{Z^+}$, which can be proved by successive Integration By Parts,  
$$ \gamma(a,b) = (a-1)! \left( 1- e^{-b} \sum_{r=0}^{a-1} \dfrac{b^r}{r!} \right) \ \forall \ a \in \mathbb{Z^+} $$ 
Thus,  
$$ I_{k,N} = \Im \left( \dfrac{N!}{(N-2ik \pi x)^{N+2}} \left( 1-e^{-(N-2ik \pi x)} \sum_{r=0}^{N} \dfrac{(N-2ik \pi x)^r}{r!} \right) \right) $$
