Question
Evaluate the following integrals:
$\int\frac{\text{x}^2}{\text{x}^6+\text{a}^6}\text{dx}$
$\int\frac{\text{x}^2}{\text{x}^6+\text{a}^6}\text{dx}$
✓
Answer
$\int\frac{\text{x}^2}{\text{x}^6+\text{a}^6}\text{dx}$
$\Rightarrow\int\frac{\text{x}^2\text{dx}}{(\text{x}^3)^2+(\text{a}^3)^2}$
Let $\text{x}^3=\text{t}$
$\Rightarrow3\text{x}^3\text{dx = dt}$
$\Rightarrow\text{x}^2\text{dx}=\frac{\text{dt}}{3}$
Now $\int\frac{\text{x}^2}{\text{x}^6+\text{a}^6}\text{dx}$
$=\frac{1}{3}\int\frac{\text{dt}}{\text{t}^2+(a^3)^2}$
$=\frac{1}{3\text{a}^3}\tan^{-1}\Big(\frac{\text{t}}{\text{a}^3}\Big)+\text{C}$
$=\frac{1}{3\text{a}^3}\tan^{-1}\Big(\frac{\text{x}^3}{\text{a}^3}\Big)+\text{C}$
$\Rightarrow\int\frac{\text{x}^2\text{dx}}{(\text{x}^3)^2+(\text{a}^3)^2}$
Let $\text{x}^3=\text{t}$
$\Rightarrow3\text{x}^3\text{dx = dt}$
$\Rightarrow\text{x}^2\text{dx}=\frac{\text{dt}}{3}$
Now $\int\frac{\text{x}^2}{\text{x}^6+\text{a}^6}\text{dx}$
$=\frac{1}{3}\int\frac{\text{dt}}{\text{t}^2+(a^3)^2}$
$=\frac{1}{3\text{a}^3}\tan^{-1}\Big(\frac{\text{t}}{\text{a}^3}\Big)+\text{C}$
$=\frac{1}{3\text{a}^3}\tan^{-1}\Big(\frac{\text{x}^3}{\text{a}^3}\Big)+\text{C}$
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