Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSIndefinite Integrals2 Marks
Question
Evaluate the following integrals:$\int\text{e}^{\text{x}}\Big(\frac{\text{x}-1}{2\text{x}^2}\Big)\text{dx}$
✓
Answer
Let $\text{I}=\int\text{e}^\text{x}\frac{1}{2\text{x}}\text{dx}-\int\text{e}^{\text{x}}\frac{1}{2\text{x}^2}\text{dx}$
Integration by parts
$=\frac{\text{e}^{\text{x}}}{2\text{x}}-\int\text{e}^{\text{x}}\Big(\frac{\text{d}}{\text{dx}}\Big(\frac{1}{2\text{x}}\Big)\Big)\text{dx}-\int\frac{\text{e}^{\text{x}}}{2\text{x}^2}\text{dx}$
$=\frac{\text{e}^{\text{x}}}{2\text{x}}+\int\frac{\text{e}^{\text{x}}}{2\text{x}^2}\text{dx}-\int\frac{\text{e}^{\text{x}}}{2\text{x}^2}\text{dx}$
$=\frac{\text{e}^{\text{x}}}{2\text{x}}+\text{C}$
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