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Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals1 Mark
Question
$\int\frac{\text{x}+3}{(\text{x}+4)^2}\text{e}^{\text{x}}\text{ dx}=$
  1. $\frac{\text{e}^\text{x}}{\text{x}+4}+\text{C}$
  2. $\frac{\text{e}^\text{x}}{\text{x}+3}+\text{C}$
  3. $\frac{1}{(\text{x}+4)^2}+\text{C}$
  4. $\frac{\text{e}^\text{x}}{(\text{x}+4)^2}+\text{C}$
✓

Answer

  1. $\frac{\text{e}^\text{x}}{\text{x}+4}+\text{C}$

Solution:

$\text{I}=\int\frac{\text{x}+3}{(\text{x}+4)^2}\text{e}^{\text{x}}\text{ dx}$

$\text{I}=\int\bigg(\frac{\text{x}+4-1}{(\text{x}+4)^2}\bigg)\text{e}^{\text{x}}\text{ dx}$

$\text{I}=\int\Big(\frac{1}{\text{x}+4}-\frac{1}{(\text{x}+4)^2}\Big)\text{e}^{\text{x}}\text{ dx}$

$\text{f(x)}=\frac{1}{\text{x}+4}$

$\text{f}'(\text{x})=-\frac{1}{(\text{x}+4)^2}$

$\text{I}=\frac{\text{e}^{\text{x}}}{\text{x}+4}+\text{C}$

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