Question

Calcule a integral por partes

Answer 1

∫$\frac{x}{ e^{x} } dx$

u = x         dv = $e^{-x}$dx
du = dx      v = $-e^{-x}$

uv - ∫vdu

x . ($- e^{-x}$) - ∫-$e^{-x}dx$

$-x.e^{-x}$ - $e^{-x}$ + c

$-e^{-x} .(x+1) + c$

Answer 2

$\sf \displaystyle \int \frac{x}{e^x}dx\\\\\\=-\frac{x}{e^x}-\int \:-\frac{1}{e^x}dx\\\\\\=-\frac{x}{e^x}-\frac{1}{e^x}\\\\\\\to \boxed{\sf =-\frac{x}{e^x}-\frac{1}{e^x}+C}$