Mathematics

$$\int_{}^{} {\dfrac{{{{(x - {x^5})}^{\cfrac{1}{5}}}}}{{{x^6}}}} dx$$ is equal to


ANSWER

$$ - \dfrac{5}{{24}}{\left( {\dfrac{1}{{{x^4}}} - 1} \right)^{\cfrac{6}{5}}} + C$$


SOLUTION
$$\displaystyle\int \dfrac{(x-x^5)^{1/5}}{x^6}dx$$
$$=\displaystyle\int \dfrac{x\left(\dfrac{1}{x^4}-1\right)^{1/5}}{x^6}dx$$
$$=\displaystyle\int \dfrac{\left(\dfrac{1}{x^4}-1\right)^{1/5}}{x^5}dx$$
$$\dfrac{1}{x^4}-1=t$$
$$=\dfrac{-4}{x^5}dx=dt$$
$$\dfrac{dx}{x^5}=-\dfrac{dt}{4}$$
$$=\displaystyle\int \dfrac{-t^{1/5}dt}{4}$$
$$=\dfrac{-5}{24}t^{6/5}+c$$
$$=\dfrac{-5}{24}\left(\dfrac{1}{x^4}-1\right)^{6/5}+c$$
C is correct.
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Single Correct Medium Published on 17th 09, 2020
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