Question

$\int {\left(\frac{x}{1+x^5}\right)}^{3/2}dx$ equals to

• A. $\frac{2}{5}\sqrt{\frac{x^5}{1+x^5}}+c$
• B. $\frac{2}{5}\sqrt{\frac{x}{1+x^5}}+c$
• C. $\frac{2}{5}\sqrt{\frac{1}{1+x^5}}+c$
• D. none of these

Answer 1

The correct option is A $\frac{2}{5}\sqrt{\frac{x^5}{1+x^5}}+c$

$I=\int {\left(\frac{x}{(1+x^5)}\right)}^{3/2}dx=\int \frac{x^{-6}}{(1+x^{-5}{)}^{3/2}}dx$

Substitute $t=1+x^{-5};dt=-5x^{-6}dx$

$\therefore I=-\frac{1}{5}\int \frac{dt}{(t{)}^{3/2}}=\frac{2}{5}{t}^{-1/2}+c$

$=\frac{2}{5}(1+x^{-5}{)}^{-1/2}+c$

$=\frac{2}{5}\sqrt{\frac{x^5}{1+x^5}}+c$