Putting x^2 = t 2xdx = dt ∫dt√(1 t t^2) ∫dt√(54 (t^2 + 2t2 + 14)) ∫dt√(54 (t 12)^2) sin^ 12t+1√(5)+C sin^ 12x^2+1√(5)+C ...

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Integration of Irrational Algebraic Fractions - 2

∫2x√1 - x2 - ...

Question

$\int \frac{2 x}{\sqrt{1 - x^{2} - x^{4}}} d x$

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Solution

Putting $x^{2} = t$
$2 x d x = d t$
$\int \frac{d t}{\sqrt{1 - t - t^{2}}}$
$\int \frac{d t}{\sqrt{\frac{5}{4} - (t^{2} + 2 \frac{t}{2} + \frac{1}{4})}}$
$\int \frac{d t}{\sqrt{\frac{5}{4} - (t - \frac{1}{2})^{2}}}$
$s i n^{- 1} \frac{2 t + 1}{\sqrt{5}} + C$
$s i n^{- 1} \frac{2 x^{2} + 1}{\sqrt{5}} + C$

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