Question

integral de 0 até -∞ de e^x dx

Answer 1

Resolução da questão, veja:

Vamos utilizar a seguinte propriedade para resolver isso:

$\Large\boxed{\boxed{\boxed{\boxed{\mathbf{\displaystyle\int\limits_{a}^{\infty}f(t)~dt=\displaystyle\lim_{u~\to~\infty}\displaystyle\int\limits_{a}^{u}~f(t)~dt}}}}}}}}}}}}}}}}}}}}}}}}}}}}}$

Veja:

$\mathtt{\displaystyle\int\limits_{0}^{-\infty}~e^{x}~dx}}}}}}\\\\\\\\ \mathtt{\displaystyle\lim_{t~\to~-\infty}~\displaystyle\int\limits_{0}^{t}~e^{x}~dx}}}}}}}\\\\\\\\\ \mathtt{e^{x}\bigg|_{0}^{t}}}}}}\\\\\\\\\ \mathtt{e^{x}-e^{0}}}}}}}\\\\\\\\\ \mathtt{e^{t}-1^{-}}}}}\\\\\\\\\ \mathtt{\displaystyle\lim_{t~\to~-\infty}~e^{t}-1^{-}}}}}}}}}}\\\\\\\\\ \mathtt{e^{-\infty}-1^{-}}}}}}\\\\\\\\\ \mathtt{0-1^{-}}}}}}}}\\\\\\\\\\\ \Large\boxed{\boxed{\boxed{\mathbf{~\therefore~\displayatyle\int\limits_{0}^{-\infty}~e^{x}~dx=-1^{-}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}$

Espero que te ajude. '-'