Question

Please answer this question​

Answer 1

Simplify the integrand:

`(\left(x-x^3\right)^(\frac13))/(x^4) = \frac{\left(x^3 \left(\frac1{x^2}-1\right)\right)^(\frac13)}{x^4}= \frac{x \left(\frac1{x^2} - 1\right)^(\frac13)}{x^4} = \frac{\left(\frac1{x^2}-1\right)^(\frac13)}{x^3}`

Substitute y = 1/x² - 1 and dy = -2/x³ dx :

`\displaystyle \int \frac{\left(\frac1{x^2}-1\right)^(\frac13)}{x^3} \, dx = -\frac12 \int y^(\frac13) \, dy \\\\ = -\frac12 * \frac34 y^(\frac43) + C \\\\ = \boxed{-\frac38 \left(\frac1{x^2}-1\right)^(\frac43) + C}`

Answer 2

`\rm \int \frac{(x - {x}^3 {)}^{ (1)/(3) } }{ {x}^(4) } dx \\`

can be written as:-

`\rm\int{(- (x^(3))/(3) + (x)/(3))/(x^(4)) d x} \\`

`\rm = {\int{(1 - x^(2))/(3 x^(3)) d x}} \\`

`\rm = {\left(\frac{\displaystyle \rm\int{(1 - x^(2))/(x^(3)) d x}}{3}\right)} \\`

`= \rm \frac{{\displaystyle \rm\int{\left(- (1)/(x) + (1)/(x^(3))\right)d x}}}{3} \\`

`= \rm \frac{{\left( \displaystyle \rm\int{(1)/(x^(3)) d x} - \int{(1)/(x) d x}\right)}}{3} \\`

`\rm\int{(- (x^(3))/(3) + (x)/(3))/(x^(4)) d x} = - \frac{\ln{\left(\left|{x}\right| \right)}}{3} - (1)/(6 x^(2))+C \\`