Question

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

Answer 1

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

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

$=\int \frac{x(1-\frac{1}{x^3}{)}^{1/4}}{x^5}.dx$

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

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

$-\left(\frac{-3}{x^4}\right)=\frac{dt}{dx}$

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

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

$\to =\int t^{2/4}\left(\frac{dt}{3}\right)$

$=\frac{1}{3}\int t^{\frac{1}{4}}.dt$

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

$=\frac{1\times 4}{3\times 5}{t}^{5/4}+C$

$=\frac{4}{15}{t}^{5/4}+C$

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