$$ \begin{array}{l} ext { The integral in this exercise converges. Evaluate the integral without using a table. } \ \int_{-\infty}^{0} heta e^{ heta} \mathrm{d} heta \ 0\end{array} e^{ heta} \mathrm{d} heta=\square $$ (Type an exact answer.)

Question

$$ \begin{array}{l}	ext { The integral in this exercise converges. Evaluate the integral without using a table. } \ \int_{-\infty}^{0} 	heta e^{	heta} \mathrm{d} 	heta \ 0\end{array} e^{	heta} \mathrm{d} 	heta=\square $$ (Type an exact answer.)

UpStudy AI · Accepted Answer

Calculate the integral $$ \int_{-\infty}^{0} 	heta e^{	heta} d	heta $$.
Evaluate the integral by following steps:
- step0: Evaluate using partial integration formula:
  $$\int_{-\infty}^{0} 	heta e^{	heta } d	heta$$
- step1: Rewrite the improper integral:
  $$\lim _{aightarrow -\infty}\left(\int_{a}^{0} 	heta e^{	heta } d	hetaight)$$
- step2: Evaluate the integral:
  $$\lim _{aightarrow -\infty}\left(-1-ae^{a}+e^{a}ight)$$
- step3: Rewrite the expression:
  $$\lim _{aightarrow -\infty}\left(-1ight)+\lim _{aightarrow -\infty}\left(-ae^{a}ight)+\lim _{aightarrow -\infty}\left(e^{a}ight)$$
- step4: Calculate:
  $$-1+0+0$$
- step5: Calculate:
  $$-1+0$$
- step6: Calculate:
  $$-1$$
The integral $$ \int_{-\infty}^{0} 	heta e^{	heta} d	heta $$ evaluates to -1.
The integral $$ \int_{-\infty}^{0} 	heta e^{	heta} d	heta $$ equals -1.

(Type an exact answer.)

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