Question

what is the answer of this definite integral​

Answer 1

Step-by-step explanation:

We have,

$\int \limits ^{1}_{ \frac{1}{3} } \frac{(x - {x}^{3} )^{ \frac{1}{3} }}{ {x}^{4} } dx$

$= \int \limits ^{1}_{ \frac{1}{3} } \frac{x(x ^{ - 2} - 1)^{ \frac{1}{3} }}{ {x}^{4} } dx$

$= \int \limits ^{1}_{ \frac{1}{3} } (x ^{ - 2} - 1)^{ \frac{1}{3} } {x}^{ - 3} dx$

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

$= \int \limits ^{0}_{ - \frac{2}{ \sqrt[3]{9} } }t .( - \frac{3}{2}) {t}^{2} dt$

$= - \frac{3}{2} \int \limits ^{0}_{ - \frac{2}{ \sqrt[3]{9} } } {t}^{3} dt$

$= - \frac{3}{8} ( {t}^{4} )^{0} _{ - \frac{2}{ \sqrt[3]{9} } }$