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Indefinite Integrals — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSIndefinite Integrals1 Mark
Question
$\int(\text{x}-1)\text{e}^{-\text{x}}\text{ dx}$ is equal to:
  1. $-\text{x}\text{e}^{\text{x}}+\text{C}$
  2. $\text{x}\text{e}^{\text{x}}+\text{C}$
  3. $-\text{x}\text{e}^{-\text{x}}+\text{C}$
  4. $\text{x}\text{e}^{-\text{x}}+\text{C}$
✓

Answer

  1. $-\text{x}\text{e}^{\text{x}}+\text{C}$
Solution:
$\int(\text{x}-1)\text{e}^{-\text{x}}\text{ dx}$
$=(\text{x}-1)\int\text{e}^{-\text{x}}\text{ dx}-\int\Big(\Big[\frac{\text{d}(\text{x}-1)}{\text{dx}}\Big]\int\text{e}^{-\text{x}}\text{dx}\Big)\text{ dx}$
$=(\text{x}-1)\frac{\text{e}^{-\text{x}}}{-1}-\int\frac{\text{e}^{-\text{x}}}{-1}\text{ dx}$
$=-(\text{x}-1)\text{e}^{-\text{x}}+\frac{\text{e}^{-\text{x}}}{-1}+\text{C}$
$=-\text{x}\text{e}^{-\text{x}}+\text{e}^{-\text{x}}+\text{C}$
$=-\text{x}\text{e}^{-\text{x}}+\text{C}$

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