Question

X^5/(1-x^3) ^1/3 integration

Answer 1

Step-by-step explanation:

We have ,

$\int \frac{ {x}^{5} }{(1 - {x}^{3})^{ \frac{1}{3} } } dx$

$Let \: \: 1 - {x}^{3} = {t}^{3} \\ \implies - 3 {x}^{2} dx = 3 {t}^{2} dt \\ \implies {x}^{2} dx = - {t}^{2} dt \: \: \: \: \:$

so,

$\int \frac{ {x}^{3} . {x}^{2} dx}{(1 - {x}^{3})^{ \frac{1}{3} } }$

$= - \int \frac{ (1 - {t}^{3} ). {t}^{2} dt}{( {t}^{3})^{ \frac{1}{3} } }$

$= - \int (1 - {t}^{3} )t dt$

$= - \int (t - {t}^{4} ) dt$

$= - \frac{ {t}^{2} }{2} + \frac{ {t}^{5} }{5} + c$

$= - {t}^{2}( \frac{ 1 }{2} + \frac{ {t}^{3} }{5} ) + c$

$= - {(1 - {x}^{3}) }^{ \frac{2}{3} }( \frac{ 1 }{2} + \frac{1 - {x}^{3} }{5} ) + c$