Question

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

Answer 1

$\int \frac{(x^4-x{)}^{3/4}}{x^5}dx$

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

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

let $1-\frac{1}{x^3}=t$

$3\times \frac{1}{x^4}dx=dt$

$\frac{1}{x^4}\times dx=\frac{dt}{3}$

$\int (t{)}^{1/4}\times \frac{dt}{3}$

$\frac{1}{3}\times \frac{1}{\left(\frac{5}{4}\right)}\times (t{)}^{5/4}$

$\frac{4}{15}\times (t{)}^{5/4}$

$\frac{4}{15}\times (1-\frac{1}{d^3}{)}^{5/4}$