Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

Question Number 181403 by a.lgnaoui last updated on 24/Nov/22

$$Referto\mathrm{Q}181319$$$$\int \frac{x^4+1}{x^4+x+2}dx$$$$\frac{x^4+1}{x^4+x+2}=1+\frac{x}{2(x^2+x+1)}-\frac{x}{2(x^2-x+1)}$$$$=1+\frac{1}{4}[\frac{(2x+1)-1}{x^2+x+1}-\frac{1}{4}\times \frac{(2x-1)+1}{(x^2-x+1)}$$$$=\frac{1}{4}\times {\left[\log ({\mathrm{x}}^2+\mathrm{x}+1)\right]}^{{}^{\prime }}-\frac{1}{4}\times \left(\frac{1}{\left[\frac{\sqrt{3}}{2}{(x+\frac{1}{2})}^2\right]+1}\right)$$$$-(\frac{1}{4}{\left[\log ({\mathrm{x}}^2-\mathrm{x}+1)\right]}^{{}^{\prime }}+\frac{1}{4}\left(\frac{1}{[\frac{\sqrt{3}}{2}\times {(x-\frac{1}{2}]}^2+1}\right))$$$$firstwith\left[\frac{\sqrt{3}}{2}(x+\frac{1}{2})\right]=u\int \frac{1}{u^2+1}du=arctg\left(u\right)$$$$andv=\left[\frac{\sqrt{3}}{2}(x-\frac{1}{2})\right]=\mathrm{v}\int \frac{dv}{1+v^2}=arctg\left(v\right)$$$$...............$$

Answered by ARUNG_Brandon_MBU last updated on 24/Nov/22

$$I=\int \frac{x^4+1}{x^4+x^2+1}dx=\int \frac{(x^4+x^2+1)-x^2}{x^4+x^2+1}dx$$$$=\int (1-\frac{x^2}{x^4+x^2+1})dx=x-\int \frac{x^2}{x^4+x^2+1}dx$$$$=x-\frac{1}{2}\int \frac{(x^2+1)+(x^2-1)}{x^4+x^2+1}dx$$$$=x-\frac{1}{2}\int \frac{x^2+1}{x^4+x^2+1}dx-\frac{1}{2}\int \frac{x^2-1}{x^4+x^2+1}dx$$$$=x-\frac{1}{2}\int \frac{1+\frac{1}{x^2}}{x^2+1+\frac{1}{x^2}}dx-\frac{1}{2}\int \frac{1-\frac{1}{x^2}}{x^2+1+\frac{1}{x^2}}dx$$$$=x-\frac{1}{2}\int \frac{1+\frac{1}{x^2}}{{(x-\frac{1}{x})}^2+3}dx-\frac{1}{2}\int \frac{1-\frac{1}{x^2}}{{(x+\frac{1}{x})}^2-1}dx$$$$=x-\frac{1}{2\sqrt{3}}\arctan \left[\frac{1}{\sqrt{3}}(x-\frac{1}{x})\right]+\frac{1}{2}\mathrm{arcoth}(x+\frac{1}{x})+C$$$$=x-\frac{\sqrt{3}}{6}\arctan \left(\frac{x^2-1}{x\sqrt{3}}\right)+\frac{1}{4}\ln \mid \frac{x^2+x+1}{x^2-x+1}\mid +C$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com