Question

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

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

Answer 1

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

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

$\int \frac{x^6}{(x+x^7)}dx$

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

$=\int \frac{(x+x^7-x)}{x(x+x^7)}dx$

$=\int \frac{x+x^7}{x(x+x^7)}dx$

$=\int \frac{x}{x(x+x^7)}dx$

$=ln|x|-\int \frac{dx}{x+x^7}$

$=ln|x|-p\left(x\right)+c$.