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integrate x*sinxdx
∫xsinx dx = x∫sinx - ∫d/dx(x)(∫sinxdx) dx .........(taking x as first function and sinx as second);
= x(-cosx) - ∫d/dx(x) (-cosx) dx
= -xcosx + ∫cosx dx
= -xcosx + sinx + C
hence ∫xsinx dx = -xcosx + sinx + C
or you can write
∫xsinx dx = Im ∫x e^(ix) dx
= Im (x∫e^(ix) dx - ∫d/dx(x) ∫e^(ix)dx dx)
= Im (xe^(ix)/i - ∫e^(ix)/i dx)
= Im (-ixe^(ix) + e^(ix)/i)
= Im (-ixe^(ix) - ie^(ix))
= Im (-ie^(ix) (x-1))
= Im (-i(cosx + isinx)(x-1))
= Im ((x-1)sinx -i(x-1)cosx)
= (x-1)sinx
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