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INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS5 Marks
Question
Evaluate the following intregals:
$\int\frac{\text{x}^2+\text{x}+1}{\text{x}^2-1}\text{ dx}\int\frac{\text{x}^2+\text{x}+1}{\text{x}^2-1}\ \text{dx}$
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Answer

Let $\text{I}=\int\frac{\text{x}^3+\text{x}+1}{\text{x}^2-1}\ \text{dx}$
$=\int\Big(\text{x}+\frac{2\text{x}+1}{\text{x}^2-1}\Big)\text{dx}$
Now,
$\frac{2\text{x}+1}{\text{x}^2-1}=\frac{\text{A}}{\text{x}+1}+\frac{\text{B}}{\text{x}-1}$
$\Rightarrow2\text{x}+1=\text{A}(\text{x}-1)+\text{B}(\text{x}+1)$
put x = 1
$\Rightarrow3=2\text{B}\Rightarrow\text{B}=\frac{3}{2}$
put x = -1
$\Rightarrow-1=-2\text{A}\Rightarrow\text{A}=\frac{1}{2}$
$\therefore\text{I}=\int\text{xdx}+\frac{1}{2}\int\frac{\text{dx}}{\text{x}+1}+\frac{3}{2}\int\frac{\text{dx}}{\text{x}-1}$
$\text{I}=\frac{\text{x}^2}{2}+\frac{1}{2}\log|\text{x}+1|+\frac{3}{2}\log|\text{x}-1|+\text{C}$

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