Question

$\int \frac{x^{3}dx}{1+x^4}$ equals

• A. $\log (x^4+1)+C$
• B. $\frac{1}{4}\log (x^4+1)+C$
• C. $\frac{1}{2}\log (x^4+1)+C$
• D. None of these

Answer 1

Answer: (B)

Let $I=\int \frac{x^3}{1+x^4}dx...(i)$
Again, let $1+x^4=t$
$\Rightarrow 4x^{3}dx=dt$
$\Rightarrow x^{3}dx=\frac{1}{4}dt$
On putting these values in Eq. $(i)$, we get
$I=\frac{1}{4}\int \frac{dt}{t}$
$=\frac{1}{4}logt+C$
$\Rightarrow I=\frac{1}{4}log(1+x^2)+C$ [From Eq. $(i)$]