Question

\[\int\left( x - 1 \right) e^{- x} dx\] is equal to

Options

• − xe^x + C

• xe^x + C

• − xe^−x + C

• xe^−x + C

Answer 1

− xe^−x + C

 

\[\int \left( x - 1 \right)_I {e^{- x}}_{II} dx\]
\[ = \left( x - 1 \right)\int e^{- x} dx - \int\left\{ \frac{d}{dx}\left( x - 1 \right)\int e^{- x} dx \right\}dx\]
\[ = \left( x - 1 \right) \cdot e^{- x} \left( - 1 \right) - \int1 \cdot e^{- x} \times - 1 dx\]
\[ = - \left( x - 1 \right) e^{- x} + \int e^{- x} dx\]
\[ = \left( 1 - x \right) e^{- x} + \frac{e^{- x}}{- 1} + C\]
\[ \Rightarrow \left( 1 - x - 1 \right) e^{- x} + C\]
\[ = - x e^{- x} + C\]