Integrate $\int \frac{x^4+1}{(x^4+2)^{3/4}}dx$

Question

This is a question of indefinite integration.

I tried many substitutions but could not solve it. It is to be integrated from $0$ to $1$. $t=x^4+1$ can't be applied here.

Answer 1

Let $u=x^4+1$, then $d u=4 x^3 d x=4(u-1)^{\frac{3}{4}} d x$ $$ \begin{aligned} \int \frac{x^4+1}{\left(x^4+2\right)^{\frac{3}{4}}} d x &\stackrel{\phantom{s=u^2-1}}{=}\int \frac{u}{(u+1)^{\frac{3}{4}}} \cdot \frac{d u}{4(u-1)^{\frac{3}{4}}} \\ &\stackrel{\phantom{s=u^2-1}}{=}\frac{1}{4} \int \frac{u d u}{\left(u^2-1\right)^{\frac{3}{4}}} \\ &\stackrel{s=u^2-1}{=} \frac{1}{8} \int \frac{d s}{s^{\frac{3}{4}}}\\ &\stackrel{\phantom{s=u^2-1}}{=}\frac{1}{2} s^{\frac{1}{4}}+C \\ &\stackrel{\phantom{s=u^2-1}}{=}\frac{1}{2}(u+1)^{\frac{1}{4}}(u-1)^{\frac{1}{4}}+C \\ &\stackrel{\phantom{s=u^2-1}}{=}\frac{1}{2} x\sqrt[4]{x^4+2}+C \end{aligned} $$