Question

The integral $\int \frac{2x^3-1}{x^4+x}dx$ is equal to : (Here $C$ is a constant of integration)

• A. ${\log }_e\left|\frac{x^3+1}{x}\right|+C$
• B. $\frac{1}{2}{\log }_e\frac{(x^3+1{)}^2}{\left|x^3\right|}+C$
• C. ${\log }_e\frac{|x^3+1|}{x^2}+C$
• D. $\frac{1}{2}{\log }_e\frac{|x^3+1|}{x^2}+C$

Answer 1

The correct option is A ${\log }_e\left|\frac{x^3+1}{x}\right|+C$

$\int \frac{2x^3-1}{x^4+x}dx=\int \frac{2x-x^{-2}}{x^2+x^{-1}}dx$
Let $x^2+x^{-1}=t\Rightarrow 2x-x^{-2}=dt$
$\Rightarrow \int \frac{1}{t}dt=\ln |x^2+\frac{1}{x}|+C=\ln \left|\frac{x^3+1}{x}\right|+C$