Mathematics

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


ANSWER

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


SOLUTION
$$LHS=\displaystyle\int \left(\dfrac{x}{1+x^{5}}\right)^{3/2}dx$$
This can be re-written as
$$\displaystyle\int \dfrac{x^{4}}{(1+x^{5})^{2}}\left(\dfrac{1+x^{5}}{x^{5}}\right)^{1/2}dx$$
$$\dfrac{2}{5}\displaystyle\int \dfrac{1}{2}\dfrac{5x^{4}}{(1+x^{5})^{2}}\left(\dfrac{1+x^{5}}{x^{5}}\right)^{1/2}dx$$
$$\dfrac{2}{5}\displaystyle\int \dfrac{5x^{4}+5x^{9}-5x^{9}}{(1+x^{5})^{2}}\left(\dfrac{1+x^{5}}{x^{5}}\right)^{1/2}dx$$
$$\dfrac{2}{5}\displaystyle\int \dfrac{1}{2}\left[\dfrac{5x^{4}(1+x^{5})-5x^{4}(x^{5})}{(1+x^{5})^{2}}\right]\left(\dfrac{1+x^{5}}{x^{5}}\right)^{1/2}dx$$
This is $$d\left(\dfrac{x^{5}}{1+x^{5}}\right)$$
$$\Rightarrow \dfrac{2}{5}\displaystyle\int \dfrac{1}{2}\left(\dfrac{1+x^{5}}{x^{5}}\right)^{1/2}d\left(\dfrac{x^{5}}{1+x^{5}}\right)$$
let $$\dfrac{x^{5}}{1+x^{5}}=t$$
then $$\dfrac{2}{5}\displaystyle\int\dfrac{1}{2}\left(\dfrac{1}{t}\right)^{1/2}dt$$
$$=\dfrac{2}{5}\displaystyle\int \dfrac{1}{2} t^{-1/2}dt$$
$$=\dfrac{2}{5}t^{1/2}+C=\dfrac{2}{5}\sqrt{\dfrac{x^{5}}{1+x^{5}}}+C$$
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Single Correct Medium Published on 17th 09, 2020
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