Question

Evaluate: $\int \frac{dx}{(1+x^4{)}^{\frac{1}{4}}}$.

Answer 1

GE: $\int \frac{dx}{{(1+x^4)}^{1/4}}$

$\frac{\sqrt[4]{4x^4+1}}{x}=m$

$dx=\frac{dm}{\frac{x^2}{{(x^4+1)}^{3/4}}-\frac{4\sqrt{x^4+1}}{x^2}}=\frac{xdm}{[\frac{1}{m^3}-m]}$

$\Rightarrow \int \frac{1}{x}\frac{xdx}{{(1+x^4)}^{1/4}}$

$\Rightarrow \int \frac{1}{x}\frac{xdm}{[\frac{1}{m^3}-m]m}=\int \frac{-m^{2}dm}{(m^4-1)}$

$\Rightarrow \int \frac{m^2}{(m^2+1)(m+1)(m-1)}$

Partial Fraction

$\Rightarrow \int [\frac{1}{2(m^2+1)}-\frac{1}{4(m+1)}+\frac{1}{4(m-1)}]dm$

$\Rightarrow \frac{1}{2}{tan}^{-1}m-\frac{1}{4}ln(m+1)+\frac{1}{4}ln(m-1)+c$

$=\frac{1}{2}{tan}^{-1}\left|\frac{{(x^4+1)}^{1/4}}{x}\right|-\frac{1}{4}ln|\frac{{(x^4+1)}^{1/4}}{x}+1|+\frac{1}{4}ln|\frac{{(x^4+1)}^{1/4}}{x}-1|+c$