Question

$\int \frac{(x-x^5{)}^{\frac{1}{5}}}{x^6}dx$ is equal to
(where $C$ is integration constant)

• A. $\frac{5}{24}{(\frac{1}{x^4}-1)}^{\frac{6}{5}}+C$
• B. $\frac{-5}{24}{(\frac{1}{x^4}-1)}^{\frac{6}{5}}+C$
• C. $\frac{5}{24}{(1-\frac{1}{x^4})}^{\frac{6}{5}}+C$
• D. $\frac{5}{24}{(1+\frac{1}{x^4})}^{\frac{6}{5}}+C$

Answer 1

The correct option is B $\frac{-5}{24}{(\frac{1}{x^4}-1)}^{\frac{6}{5}}+C$

$I=\int \frac{(x-x^5{)}^{\frac{1}{5}}}{x^6}dx\Rightarrow I=\int \frac{x{(x^{-4}-1)}^{\frac{1}{5}}}{x^6}dx$

Put $\frac{1}{x^4}-1=t$
$\Rightarrow -\frac{4}{x^5}dx=dt\Rightarrow I=-\frac{1}{4}\int t^{\frac{1}{5}}dt\Rightarrow I=\frac{-5}{24}{t}^{\frac{6}{5}}+C\Rightarrow I=\frac{-5}{24}{(\frac{1}{x^4}-1)}^{\frac{6}{5}}+C$