Evaluate the following integrals: x 4-x^4 dx - Vidyadip

Question

Evaluate the following integrals:

$\int\frac{\text{x}}{\sqrt{4-\text{x}^4}}\text{ dx}$

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Answer

$\int\frac{\text{x}\text{ dx}}{\sqrt{4-\text{x}^4}}$
$\Rightarrow\int\frac{\text{x}\text{ dx}}{\sqrt{2^2-(\text{x}^2)^2}}$
Let $\text{x}^2=\text{t}$
$\Rightarrow2\text{x}\text{ dx}=\text{dt}$
$\text{x}\text{ dx}=\frac{\text{dt}}{2}$
Now, $\int\frac{\text{x}\text{ dx}}{\sqrt{2^2-(\text{x}^2)^2}}$
$\frac{1}{2}\int\frac{\text{dt}}{\sqrt{2^2-\text{t}^2}}$
$=\frac{1}{2}\times\sin^{-1}\Big(\frac{1}{2}\Big)+\text{C}$
$=\frac{1}{2}\times\sin^{-1}\Big(\frac{\text{x}^2}{2}\Big)+\text{C}$

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