So I was bored, and decided to look back at some of my old questions (including my deleted questions). After a while, I found this question of mine (not deleted), and decided to create this integral problem based off it$$\int_0^2(xe^a+ax)dx=0$$which I thought that I might be able to solve. Here is my attempt at doing so:$$\int_0^2(xe^a+ax)dx=0\implies2e^a+2a-1=0$$and now to solve our non-algebraic boi:$$2e^a+2a-1=0\implies e^a+a-0.5=0\\\implies e^a=-a+0.5\\\implies1=(-a+0.5)e^{-a}\\\implies e^{1/2}=(-a+0.5)e^{-a+0.5}\\\implies W(e^{1/2})=-a+0.5\\\implies a=0.5-W(e^{1/2})\\\approx-6.38905$$
My question
Did I solve this non-algebraic boi correctly, or what could I do to solve it correctly?
Also note that the reason we use the main branch of Lambert's $W$ function here is because we have a real solution to this problem.
$$a=-W(1)$$ $$(e^a+a)\int_0^2xdx=(e^a+a)(\frac{x^2}{2}) \vert_0^2 $$ $$2e^a+2a=0$$ $$e^a+a=0$$ $$e^a=-a$$ $$1=-ae^{-a}$$ $$a=-W(1)$$