Evaluate the following intregals: 1 x(x^4+1) dx - Vidyadip

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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