Integration Questions

Question Number 166307 by ArielVyny last updated on 18/Feb/22

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{\mathrm{4}} \left(\mathrm{1}−{x}\right)^{\mathrm{4}} }{\mathrm{1}+{x}^{\mathrm{2}} }{dx} \\$$

Answered by MJS_new last updated on 18/Feb/22

$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\frac{{x}^{\mathrm{4}} \left(\mathrm{1}−{x}\right)^{\mathrm{4}} }{{x}^{\mathrm{2}} +\mathrm{1}}{dx}= \\$$ $$=\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\left({x}^{\mathrm{6}} −\mathrm{4}{x}^{\mathrm{5}} +\mathrm{5}{x}^{\mathrm{4}} −\mathrm{4}{x}^{\mathrm{2}} +\mathrm{4}−\frac{\mathrm{4}}{{x}^{\mathrm{2}} +\mathrm{1}}\right){dx}= \\$$ $$=\left[\frac{{x}^{\mathrm{7}} }{\mathrm{7}}−\frac{\mathrm{2}{x}^{\mathrm{6}} }{\mathrm{3}}+{x}^{\mathrm{5}} −\frac{\mathrm{4}{x}^{\mathrm{3}} }{\mathrm{3}}+\mathrm{4}{x}−\mathrm{4arctan}\:{x}\right]_{\mathrm{0}} ^{\mathrm{1}} = \\$$ $$=\frac{\mathrm{22}}{\mathrm{7}}−\pi \\$$

Commented byArielVyny last updated on 19/Feb/22

$${exact}\:{sir}\:{thank} \\$$