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Question Number 167036 by HongKing last updated on 05/Mar/22

Evaluate:  Ω = ∫ 620 (x^(2017)  - 69 x^(126) )^(15)  dx

$$\mathrm{Evaluate}: \\ $$$$\Omega\:=\:\int\:\mathrm{620}\:\left(\mathrm{x}^{\mathrm{2017}} \:-\:\mathrm{69}\:\mathrm{x}^{\mathrm{126}} \right)^{\mathrm{15}} \:\mathrm{dx} \\ $$

Answered by mr W last updated on 05/Mar/22

Ω = ∫ 620 (x^(2017)  - 69 x^(126) )^(15)  dx  =620 ∫x^(2017×15)  (1 - 69x^(−1891) )^(15)  dx  =620 ∫Σ_(k=0) ^(15) C_k ^(15) (−1)^k 69^k x^(30255−1891k)  dx  =620Σ_(k=0) ^(15) (((−1)^k C_k ^(15) 69^k x^(30256−1891k) )/(30256−1891k))+C

$$\Omega\:=\:\int\:\mathrm{620}\:\left(\mathrm{x}^{\mathrm{2017}} \:-\:\mathrm{69}\:\mathrm{x}^{\mathrm{126}} \right)^{\mathrm{15}} \:\mathrm{dx} \\ $$$$=\mathrm{620}\:\int{x}^{\mathrm{2017}×\mathrm{15}} \:\left(\mathrm{1}\:-\:\mathrm{69}{x}^{−\mathrm{1891}} \right)^{\mathrm{15}} \:\mathrm{dx} \\ $$$$=\mathrm{620}\:\int\underset{{k}=\mathrm{0}} {\overset{\mathrm{15}} {\sum}}{C}_{{k}} ^{\mathrm{15}} \left(−\mathrm{1}\right)^{{k}} \mathrm{69}^{{k}} {x}^{\mathrm{30255}−\mathrm{1891}{k}} \:\mathrm{dx} \\ $$$$=\mathrm{620}\underset{{k}=\mathrm{0}} {\overset{\mathrm{15}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}} {C}_{{k}} ^{\mathrm{15}} \mathrm{69}^{{k}} {x}^{\mathrm{30256}−\mathrm{1891}{k}} }{\mathrm{30256}−\mathrm{1891}{k}}+{C} \\ $$

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