Question

If $\int \frac{dx}{x+x^7}=p\left(x\right),$ then $\int \frac{x^6}{x+x^7}dx$ is equal to
(where $C$ is constant of integration)

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

Answer 1

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

Let
$I=\int \frac{x^6}{x+x^7}dx$
$\Rightarrow I=\int \frac{x^6}{x(1+x^6)}dx$
$\Rightarrow I=\int \frac{(1+x^6)-1}{x(1+x^6)}dx$
$\Rightarrow I=\int \frac{1}{x}dx-\int \frac{1}{x+x^7}dx$
$\therefore I=\ln |x|-p\left(x\right)+C$