جواب سوال :
$$
\int \frac{x^{4}+2}{x^{4}+x^{3}+x^{2}}=\int \frac{x^{2}\left(x^{2}+\frac{2}{x^{2}}\right)}{x^{2}\left(x^{2}+x+1\right)}=\underbrace{\int \frac{x^{2}}{\left(x^{2}+x+1\right)}}_{A}+\underbrace{\int \frac{2}{x^{2}\left(x^{2}+x+1\right)}}_{B}=A+B \\
A=\int \frac{x^{2}}{\left(x^{2}+x+1\right)}=\int \frac{x^{2}+x+1-x-1}{\left(x^{2}+x+1\right)}=x-\int \frac{x+1}{\left(x^{2}+x+1\right)}=x-\frac{1}{2} \int \frac{2\left(x+\frac{1}{2}\right)}{\left(x^{2}+x+1\right)}-\frac{1}{2} \int \frac{1}{\left(x^{2}+x+1\right)} \\
=x-\frac{1}{2} \ln \left(x^{2}+x+1\right)-\frac{1}{\sqrt{3}} A r c \tan \left(\frac{x+\frac{1}{2}}{\sqrt{3}}\right) \frac{\left(x+\frac{1}{2}\right)}{\sqrt{3}} \\
B=\int \frac{2}{x^{2}\left(x^{2}+x+1\right)}=\int \frac{-2 x+2}{x^{2}}+\int \frac{2 x}{x^{2}+x+1}=-2 \ln |x|-\frac{2}{x}+\ln \left(x^{2}+x+1\right)+\frac{2}{\sqrt{3}} A r c \tan \left(\frac{x+\frac{1}{2}}{\frac{\sqrt{3}}{2}}\right)
$$
