Question

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

Answer 1

Now,

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

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

Let, $1-\frac{1}{x^3}=z$ then $\frac{3}{x^4}dx=dz$

Using these in the above expression we get,

$=\int \frac{z^{1/4}}{3}dz$

$=\frac{4}{15}{z}^{5/4}+c$ [ Where $c$ is integrating constant]

$=\frac{4}{15}(1-\frac{1}{x^3}{)}^{5/4}+c$