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}}{\left|x^{3}\right|} +C $
• C. $\frac{1}{2} \log_{e} \frac{\left|x^{3} +1\right|^{2}}{x^{3}} +C $
• D. $ \log_{e} \frac{\left|x^{3} +1\right|^{2}}{x^{3}} +C $

Answer 1

Answer: (A)

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