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Question Number 219458 by Nicholas666 last updated on 25/Apr/25

∫_1 ^( 2) ∫_1 ^( 2) ∫_1 ^( 2) ∫_1 ^( 2) x_1 ^( 2) x_2 ^( 3) x_3 ^( 4) x_4 ^( 5) dx_1 dx_2 dx_3 dx_(4 )

$$ \\ $$$$\:\int_{\mathrm{1}} ^{\:\mathrm{2}} \int_{\mathrm{1}} ^{\:\mathrm{2}} \int_{\mathrm{1}} ^{\:\mathrm{2}} \int_{\mathrm{1}} ^{\:\mathrm{2}} \:{x}_{\mathrm{1}} ^{\:\mathrm{2}} {x}_{\mathrm{2}} ^{\:\mathrm{3}} {x}_{\mathrm{3}} ^{\:\mathrm{4}} {x}_{\mathrm{4}} ^{\:\mathrm{5}} \:{dx}_{\mathrm{1}} {dx}_{\mathrm{2}} {dx}_{\mathrm{3}} {dx}_{\mathrm{4}\:} \\ $$$$\: \\ $$

Answered by mr W last updated on 26/Apr/25

=(∫_1 ^2 x_1 ^2 dx_1 )(∫_1 ^2 x_2 ^3 dx_2 )(∫_1 ^2 x_3 ^4 dx_3 )(∫_1 ^2 x_4 ^5 dx_4 ) =((2^3 −1^3 )/3)×((2^4 −1^4 )/4)×((2^5 −1^5 )/5)×((2^6 −1^6 )/6) =((4557)/8)

$$=\left(\int_{\mathrm{1}} ^{\mathrm{2}} {x}_{\mathrm{1}} ^{\mathrm{2}} {dx}_{\mathrm{1}} \right)\left(\int_{\mathrm{1}} ^{\mathrm{2}} {x}_{\mathrm{2}} ^{\mathrm{3}} {dx}_{\mathrm{2}} \right)\left(\int_{\mathrm{1}} ^{\mathrm{2}} {x}_{\mathrm{3}} ^{\mathrm{4}} {dx}_{\mathrm{3}} \right)\left(\int_{\mathrm{1}} ^{\mathrm{2}} {x}_{\mathrm{4}} ^{\mathrm{5}} {dx}_{\mathrm{4}} \right) \\ $$$$=\frac{\mathrm{2}^{\mathrm{3}} −\mathrm{1}^{\mathrm{3}} }{\mathrm{3}}×\frac{\mathrm{2}^{\mathrm{4}} −\mathrm{1}^{\mathrm{4}} }{\mathrm{4}}×\frac{\mathrm{2}^{\mathrm{5}} −\mathrm{1}^{\mathrm{5}} }{\mathrm{5}}×\frac{\mathrm{2}^{\mathrm{6}} −\mathrm{1}^{\mathrm{6}} }{\mathrm{6}} \\ $$$$=\frac{\mathrm{4557}}{\mathrm{8}} \\ $$

Commented by Nicholas666 last updated on 26/Apr/25

tanks

$${tanks} \\ $$

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