Question

Question:-

$find \: \: \int \frac{ ({x}^{4 } - x ) ^{ \frac{1}{4} } } { {x}^{5} } \: dx$
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Answer 1

Question:

$find \: \: \int \frac{ ({x}^{4 } - x ) ^{ \frac{1}{4} } } { {x}^{5} } \: dx$

Solution:

$we \: have \: \int \frac{ ({x}^{4 } - x ) ^{ \frac{1}{4} } } { {x}^{5} } \: dx = \int \frac{(1 - \frac{1}{ {x}^{3} } )^{ \frac{1}{4} } }{ {x}^{4} } = dx \\ \\ put \: \: 1 - \frac{1}{ {x}^{3} } = 1 - {x}^{ - 3} = t, \: so \: \: that \: \: \frac{3}{ {x}^{4} } \: dx = dt \\ \\ therefore \: \int \frac{ ({x}^{4} - x)^{ \frac{1}{4} } } { {x}^{5} } \: dx = \frac{1}{3} \int {t}^{ \frac{1}{4} } \: dt = \frac{1}{3} \times \frac{4}{5} t ^{ \frac{5}{4} } + c = \frac{4}{5} (1 - \frac{1}{ {x}^{3} } )^{ \frac{5}{4} } + c$