Question

If $\int \frac{1}{x+x^5} d x=f(x)+C$, then the value of $\int \frac{x^4}{x+x^5} d x$ equals

(a) $\log x-f(x)+C$

(b) $f(x)+\log x+C$

(c) $f(x)-\log x+C$

(d) None of these

Answer 1

$\begin{aligned}I & =\int \frac{x^4}{x+x^5} d x=\int \frac{x^4+1-1}{x+x^5} d x=\int \frac{1}{x} d x-\int \frac{1}{x^5+x} d x \\& =\log x-[f(x)-C]=\log x-f(x)+C\end{aligned}$

So, The correct option of this question will be (A).