Question

$\int \frac{1}{\left(x+x^5\right)}dx=f\left(x\right)+C$, then the value of $\int \frac{x^4}{\left(x+x^5\right)}dx$ is

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

Answer 1

The correct option is A

$\log x-f\left(x\right)+C$

Explanation for Correct answer:

Finding the integral value:

Given, $\int \frac{1}{\left(x+x^5\right)}dx=f\left(x\right)+C\to \left(i\right)$

$$\begin{gathered} \int \frac{x^4}{\left(x+x^5\right)}dx=\int \frac{x^4}{x\left(1+x^4\right)}dx \\ =\int \frac{x^4+1-1}{x\left(1+x^4\right)}dx\left[\text{add and subtract}1\text{in numerator}\right] \\ =\int \frac{x^4+1}{x\left(1+x^4\right)}-\frac{1}{x\left(1+x^4\right)}dx \\ =\int \frac{1}{x}dx-\int \frac{1}{x\left(1+x^4\right)}dx \\ =\log x-f\left(x\right)+C\left[\text{from equation}\left(i\right)\right] \end{gathered}$$

Hence, option (A) is the correct answer.