Question

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

• A. $In|x|-p\left(x\right)+c$
• B. $In|x|+p\left(x\right)+c$
• C. $x-p\left(x\right)+c$
• D. $x+p\left(x\right)+c$

Answer 1

The correct option is A $In|x|-p\left(x\right)+c$

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

Now,

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

Hence, this is the answer.