Question

$ \int \frac{x^3dx}{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} log \,t +C$
$\Rightarrow I = \frac{1}{4} log (1 + x^2) + C $ [From Eq. $(i)$]