Integration Questions

Question Number 67197 by necxxx last updated on 23/Aug/19

$$\int_{\mathrm{0}} ^{\mathrm{2}} {x}^{\mathrm{5}} \left(\mathrm{1}−\frac{{x}}{\mathrm{2}}\right)^{\mathrm{4}} {dx} \\$$

Answered by turbo msup by abdo last updated on 24/Aug/19

$${I}\:=\frac{\mathrm{1}}{\mathrm{16}}\:\int_{\mathrm{0}} ^{\mathrm{2}} {x}^{\mathrm{5}} \left({x}−\mathrm{2}\right)^{\mathrm{4}} {dx} \\$$ $$=\frac{\mathrm{1}}{\mathrm{16}}\:\int_{\mathrm{0}} ^{\mathrm{2}} {x}^{\mathrm{5}} \sum_{{k}=\mathrm{0}} ^{\mathrm{4}} \:{C}_{\mathrm{4}} ^{{k}} {x}^{{k}} \left(−\mathrm{2}\right)^{\mathrm{4}−{k}} \\$$ $$\mathrm{16}{I}=\sum_{{k}=\mathrm{0}} ^{\mathrm{4}} \:{C}_{\mathrm{4}} ^{{k}} \left(−\mathrm{2}\right)^{\mathrm{4}−{k}} \int_{\mathrm{0}} ^{\mathrm{2}} \:{x}^{{k}+\mathrm{5}} \:{dx} \\$$ $$=\mathrm{16}\:\sum_{{k}=\mathrm{0}} ^{\mathrm{4}} \:{C}_{\mathrm{4}} ^{{k}} \:\left(−\mathrm{2}\right)^{−{k}} \:\left[\frac{\mathrm{1}}{{k}+\mathrm{6}}{x}^{{k}+\mathrm{6}} \right]_{\mathrm{0}} ^{\mathrm{2}} \\$$ $$\Rightarrow{I}\:=\overset{\mathrm{4}} {\sum}_{{k}=\mathrm{0}} \left(−\mathrm{2}\right)^{−{k}} \:{C}_{\mathrm{4}} ^{{k}} \:×\frac{\mathrm{2}^{{k}+\mathrm{6}} }{{k}+\mathrm{6}} \\$$