$\int \frac{1}{x^{2} (x^{4} + 1)^{\frac{3}{4}}}$ dx is equal to __________

$\frac{- (1 + x^{4})^{\frac{1}{4}}}{x} + C$

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$\frac{- (1 + x^{4})^{\frac{1}{4}}}{x^{2}} + C$

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$\frac{- (1 + x^{4})^{\frac{1}{4}}}{2 x} + C$

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$\frac{- (1 + x^{4})^{\frac{3}{4}}}{x} + C$

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Solution

The correct option is A
$\frac{- (1 + x^{4})^{\frac{1}{4}}}{x} + C$

$\frac{1}{x^{2} (x^{4} + 1)^{\frac{3}{4}}} = \frac{1}{x^{5} (1 + \frac{1}{x^{4}})^{\frac{3}{4}}} L e t 1 + \frac{1}{x^{4}} = t \Rightarrow \frac{1}{x^{5}} d x = - \frac{1}{4} d t ∴ \int \frac{1}{x^{2} (x^{4} + 1)^{\frac{3}{4}}} d x = \frac{- 1}{4} \int \frac{1}{t^{\frac{3}{4}}} d t = \frac{- (x^{4} + 1)^{\frac{1}{4}}}{x}$

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m

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