Question

If $\int \frac{1}{x+x^5}dx=f\left(x\right)+c$, then $\int \frac{x^4}{x+x^5}dx$ is equal to

• A. $\log \left|x\right|+f\left(x\right)+c$
• B. $\log \left|x\right|-f\left(x\right)+c$
• C. $xf\left(x\right)+c$
• D. None of these

Answer 1

Answer: (B)

$\log \left|x\right|-f\left(x\right)+c$

Let $I_1=\int \frac{1}{x+x^5}dx$ and $I_2=\int \frac{x^4}{x+x^5}dx$

$I_1+I_2=\int (\frac{1}{x+x^5}+\frac{x^4}{x+x^5})dx=\int \frac{1+x^4}{x+x^5}dx$

$=\int \frac{dx}{x}=\log \left|x\right|+c$

As $I_1=f\left(x\right)+c$

$\therefore I_2=\log \left|x\right|-f\left(x\right)+c$