Mathematics

Integrate with respect to $$x$$:
$$\displaystyle \int \dfrac {1}{x^{6}{(1+x^{-5})}^{\frac {1}{5}}}dx$$


SOLUTION
We have,
$$I=\displaystyle \int \dfrac {1}{x^{6}{(1+x^{-5})}^{\frac {1}{5}}}dx$$

Let
$$t=1+x^{-5}$$

$$\dfrac{dt}{dx}=0-\dfrac{5}{x^6}$$

$$\dfrac{dt}{5}=-\dfrac{dx}{x^6}$$

Therefore,

$$I=-\dfrac{1}{5}\displaystyle \int \dfrac{1}{t^{\frac{1}{5}}}dt$$

$$I=-\dfrac{1}{5}\dfrac{t^{-\frac{1}{5}+1}}{-\frac{1}{5}+1}+C$$

$$I=-\dfrac{1}{5}\dfrac{t^{\frac{4}{5}}}{\frac{4}{5}}+C$$

$$I=-\dfrac{t^{\frac{4}{5}}}{4}+C$$

On putting the value of $$t$$, we get

$$I=-\dfrac{(1+x^{-5})^{\frac{4}{5}}}{4}+C$$

Hence, this is the answer.
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Subjective Medium Published on 17th 09, 2020
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