Mathematics

Integrate:
$$\int{\dfrac{dx}{{{x}^{5}}\left( 1+{{x}^{-4}} \right)}}$$


SOLUTION

Consider the given integral.

$$I=\int{\dfrac{dx}{{{x}^{5}}\left( 1+{{x}^{-4}} \right)}}$$

 

Let $$t={{x}^{-4}}$$

$$ \dfrac{dt}{dx}=-4{{x}^{-5}} $$

$$ -\dfrac{dt}{4}=\dfrac{dx}{{{x}^{5}}} $$

 

Therefore,

$$ I=-\dfrac{1}{4}\int{\dfrac{dt}{\left( 1+t \right)}} $$

$$ I=-\dfrac{1}{4}\ln \left( 1+t \right)+C $$

 

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

$$ I=-\dfrac{1}{4}\ln \left( 1+{{x}^{-4}} \right)+C $$

$$ I=-\dfrac{1}{4}\ln \left( 1+\dfrac{1}{{{x}^{4}}} \right)+C $$

$$ I=-\dfrac{1}{4}\ln \left( \dfrac{{{x}^{4}}+1}{{{x}^{4}}} \right)+C $$

$$ I=\dfrac{1}{4}\ln \left( \dfrac{{{x}^{4}}}{{{x}^{4}}+1} \right)+C $$

 

Hence, this is the answer.

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