Question Number 167036 by HongKing last updated on 05/Mar/22
$$\mathrm{Evaluate}: \\ $$$$\Omega\:=\:\int\:\mathrm{620}\:\left(\mathrm{x}^{\mathrm{2017}} \:-\:\mathrm{69}\:\mathrm{x}^{\mathrm{126}} \right)^{\mathrm{15}} \:\mathrm{dx} \\ $$
Answered by mr W last updated on 05/Mar/22
$$\Omega\:=\:\int\:\mathrm{620}\:\left(\mathrm{x}^{\mathrm{2017}} \:-\:\mathrm{69}\:\mathrm{x}^{\mathrm{126}} \right)^{\mathrm{15}} \:\mathrm{dx} \\ $$$$=\mathrm{620}\:\int{x}^{\mathrm{2017}×\mathrm{15}} \:\left(\mathrm{1}\:-\:\mathrm{69}{x}^{−\mathrm{1891}} \right)^{\mathrm{15}} \:\mathrm{dx} \\ $$$$=\mathrm{620}\:\int\underset{{k}=\mathrm{0}} {\overset{\mathrm{15}} {\sum}}{C}_{{k}} ^{\mathrm{15}} \left(−\mathrm{1}\right)^{{k}} \mathrm{69}^{{k}} {x}^{\mathrm{30255}−\mathrm{1891}{k}} \:\mathrm{dx} \\ $$$$=\mathrm{620}\underset{{k}=\mathrm{0}} {\overset{\mathrm{15}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}} {C}_{{k}} ^{\mathrm{15}} \mathrm{69}^{{k}} {x}^{\mathrm{30256}−\mathrm{1891}{k}} }{\mathrm{30256}−\mathrm{1891}{k}}+{C} \\ $$