Evaluate the following integrals: x^6+1 x^2+1 dx - Vidyadip

Question

Evaluate the following integrals:
$\int\frac{\text{x}^6+1}{\text{x}^2+1}\text{dx}$

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Answer

$\int\Big(\frac{\text{x}^6+1}{\text{x}^2+1}\Big)\text{dx}$
$=\int\bigg[\frac{(\text{x}^2)^3+1^3}{\text{x}^2+1}\bigg]\text{dx}$ $[\text{A}^3+\text{B}^3=(\text{A+B})(\text{A}^2-\text{AB}+\text{B}^2)]$
$=\int\frac{(\text{x}^2+1)(\text{x}^4-\text{x}^2+1)}{(\text{x}^2+1)}\text{dx}$
$=\int(\text{x}^4-\text{x}^2+1)\text{dx}$
$=\int\text{x}^4\text{dx}+\int\text{x}^2\text{dx}+\int1\text{dx}$
$=\frac{\text{x}^{4+1}}{4+1}-\frac{\text{x}^{2+1}}{2+1}+\text{x}+\text{C}$
$=\frac{\text{x}^5}{5}-\frac{\text{x}^3}{3}+\text{x}+\text{C}$

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