انتگرال $\int\frac{x^4+2}{x^4+x^3+x^2}dx$ را چگونه باید حل کنم؟

Question

با سلام لطفا راهنمایی‌ام کنید که انتگرال زیر را چگونه باید حل کنم؟ ممنون $\int_{ }^{ }\frac{x^{4}+2}{x^{4}+x^{3}+x^{2}}dx$

Answer 1

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