$\int x^{- 11} {(1 + x^{4})}^{- 1 / 2} d x is equal to$

Moderate

A

$\frac{1}{2} [\frac{t^{5}}{5} + \frac{2 t^{3}}{3} + t] + c, where t = \sqrt{1 + \frac{1}{x^{4}}}$
B

$- \frac{1}{2} [\frac{t^{5}}{5} - \frac{2 t^{3}}{3} + t] + c, where t = \sqrt{1 + \frac{1}{x^{4}}}$
C

$[\frac{t^{5}}{5} - \frac{2 t^{3}}{3} + t] + c, where t = \sqrt{1 + \frac{1}{x^{4}}}$
D

None of these

Solution

$\begin{array}{l} L e t I = \int \frac{d x}{x^{11} {(1 + x^{4})}^{1 / 2}} = \int \frac{d x}{x^{11} \cdot x^{2} {(1 + 1 / x^{4})}^{1 / 2}} \\ L e t 1 + \frac{1}{x^{4}} = t^{2} . Then \frac{- 4}{x^{5}} d x = 2 t d t \\ \Rightarrow I = \int \frac{d x}{x^{13} {(1 + 1 / x^{4})}^{1 / 2}} \\ = - \frac{1}{4} \int \frac{2 t d t}{x^{8} t} \\ = - \frac{1}{2} \int {(t^{2} - 1)}^{2} d t \\ = - \frac{1}{2} \int (t^{4} - 2 t^{2} + 1) d t \\ = - \frac{1}{2} [\frac{t^{5}}{5} - \frac{2 t^{3}}{3} + t] + c, where t = \sqrt{1 + \frac{1}{x^{4}}} \end{array}$

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