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INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS4 Marks
Question
Evaluate the following intregals:
$\int\frac{1}{\text{x}(\text{x}^4+1)}\ \text{dx}$
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Answer

Let $\frac{1}{\text{x}(\text{x}^4+1)}+\frac{\text{A}}{\text{x}}+\frac{\text{Bx}^3+\text{Cx}^2+\text{Dx}+\text{E}}{\text{x}^4+1}$

$\Rightarrow1=\text{A}(\text{x}^4+1)+(\text{Bx}^3+\text{Cx}^2+\text{Dx}+\text{E})\text{x}$

$=(\text{A}+\text{B})\text{x}^4+\text{Cx}^3+\text{Dx}^2+\text{Ex}+\text{A}$

Equating similar terms, we get,

$\text{A}+\text{B}+0,\text{C}=0,\text{E}=0,\text{A}=1$

$\therefore\text{B}=-1$

Thus,

$\text{I}=\int\frac{\text{dx}}{\text{x}}+\int-\frac{\text{x}^3\text{dx}}{\text{x}^4+1}$

$=\log|\text{x}|-\frac{1}{4}\log|\text{x}^4+1|+\text{C}$

$\text{I}=\frac{1}{4}\log\Big|\frac{\text{x}^4}{\text{x}^4+1}\Big|+\text{C}$

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