Mathematics

If $$\int \dfrac{dx}{x+x^{7}}=p(x)$$ then , $$\int \dfrac{x^{6}}{x+x^{7}}dx$$ is eqaul to: 


ANSWER

$$In |x|-p\ (x)+c$$


SOLUTION
$$\int{\cfrac{dx}{x + {x}^{7}}} = p\left( x \right) \quad \left( \text{Given} \right)$$

Now,
$$\int{\cfrac{{x}^{6}}{x + {x}^{7}} dx} \\ = \int{\cfrac{{x}^{6} + 1 - 1}{x + {x}^{7}} dx} \\ = \int{\cfrac{{x}^{6} + 1}{x + {x}^{7}} dx} - \int{\cfrac{dx}{x + {x}^{7}}} \\ = \int{\cfrac{{x}^{6} + 1}{x \left( 1  + {x}^{6} \right)} dx} - \int{\cfrac{dx}{x + {x}^{7}} dx} \\ = \ln{\left| x \right|} - p \left( x \right) + c$$

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