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INTEGRALS — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSINTEGRALS1 Mark
Question
Fill in the blanks:
$\int\frac{\text{x}+3}{(\text{x}+4)^2}\text{e}^\text{x}\text{dx}=$ ________.
✓

Answer

$\int\frac{\text{x}+3}{(\text{x}+4)^2}\text{e}^\text{x}\text{dx}=\text{e}^{\text{x}}\frac{1}{\text{x}+4}+\text{C}$
Solution:
$\text{I}=\int\frac{\text{x}+3}{(\text{x}+4)^2}\text{e}^\text{x}\text{dx}$
$=\int\frac{\text{x}+4-1}{(\text{x}+4)^2}\text{e}^\text{x}\text{dx}$
$=\int\text{e}^\text{x}\Big(\frac{1}{\text{x}+4}-\frac{1}{(\text{x}+4)^2}\Big)\text{dx}$
$=\int\text{e}^\text{x}\big(\text{f(x)}+\text{f}'(\text{x})\big)\text{dx,}$ where $=\text{e}^{\text{x}}\frac{1}{\text{x}+4}+\text{C}$

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