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

Answer 1

Answer: (A)

$\int \frac{2x^3-1}{x^4+x}dx$
$\Rightarrow \int \frac{\left(4x^3+1\right)-\left(2x^3+2\right)}{x^4+x}dx$
$\Rightarrow \int \frac{4x^3+1}{x^4+x}dx-2\int \frac{1}{x}dx$
$x^4+x=t\Rightarrow \left(4x^3+1\right)dx=dt$
$\Rightarrow \int \frac{dt}{t}-2\int \frac{1}{x}dx$
$\Rightarrow \ell n|t|-2\ell nx+C$
$\Rightarrow \ell n\left|\frac{x^4+x}{x^2}\right|+C\Rightarrow \ell n\left|\frac{x^3+1}{x}\right|+C$