Question
Evaluate the following intregals:
$\int\frac{1}{\text{x}(\text{x}^4+1)}\ \text{dx}$
$\int\frac{1}{\text{x}(\text{x}^4+1)}\ \text{dx}$
$\Rightarrow1=\text{A}(\text{x}^4+1)+(\text{Bx}^3+\text{Cx}^2+\text{Dx}+\text{E})\text{x}$
$=(\text{A}+\text{B})\text{x}^4+\text{Cx}^3+\text{Dx}^2+\text{Ex}+\text{A}$
Equating similar terms, we get,
$\text{A}+\text{B}+0,\text{C}=0,\text{E}=0,\text{A}=1$
$\therefore\text{B}=-1$
Thus,
$\text{I}=\int\frac{\text{dx}}{\text{x}}+\int-\frac{\text{x}^3\text{dx}}{\text{x}^4+1}$
$=\log|\text{x}|-\frac{1}{4}\log|\text{x}^4+1|+\text{C}$
$\text{I}=\frac{1}{4}\log\Big|\frac{\text{x}^4}{\text{x}^4+1}\Big|+\text{C}$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Maximize Z = x + y
Subject to
$-2\text{x}+\text{y}\leq1$
$\text{x}\leq2$
$\text{x}+\text{y}\leq3$
$\text{x},\text{y}\geq0$