MHB Why is the integration result not -xe^-x - e^-x?

[Cadbury]

View attachment 3778

So v' is the one to be integrated and v is the answer,
why is it not -xe^-x - e^-x ?

Thank you very much!

 

[Fantini]

Hello Cadbury. What is the result of $$\int x e^{-x} \, dx?$$

 

[Cadbury]

Hello Cadbury. What is the result of $$\int x e^{-x} \, dx?$$

Oh, I get it now haha, first I have to use integration by parts where u= x, dv= e^ -x then uv - integral(vdu) then after

-xe^-x +e^-x - e^-x = -xe^-x hehe

 

[Prove It]

It's even easier if you write the first DE as $\displaystyle \begin{align*} y' = \left( x - 1 \right) \, \mathrm{e}^{-x} \end{align*}$, you can still apply integration by parts and this time you don't have to do two separate integrals...