Mathematics

Evaluate: $$\displaystyle \int_{1/3}^{1}\frac{(x-x^{3})^{1/3}}{x^{4}}dx$$


ANSWER

$$6$$


SOLUTION

$$\displaystyle \int_{\frac {1}{3}}^{1} \dfrac {(x – x^{3})^{\frac {1}{3}}}{x^{4}} dx = \int_{\frac {1}{3}}^{1} \frac {(x^{3})^{\frac {1}{3}} \left (\dfrac {1}{x^{2}} – 1\right )^{\frac {1}{3}}}{x^{4}} dx$$

$$\displaystyle = \int_{\dfrac {1}{3}}^{1} \dfrac {\left (\dfrac {1}{x^{2}} – 1\right )^{\frac {1}{3}}}{x^{3}} dx$$           (let $$\dfrac {1}{x^{2}} – 1 = t$$)

$$\displaystyle = \int_{8}^{0} \dfrac{t^{\frac{1}{3}}}{-2} dt$$    $$\therefore \dfrac {-2}{x^{3}} dx = dt$$

$$= -\dfrac {1}{2}\left (\dfrac {t^{4/3}}{4/3}\right )_{8}^{0} = 6$$.
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Single Correct Medium Published on 17th 09, 2020
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