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Question Number 199638 by witcher3 last updated on 06/Nov/23

∫_0 ^1 ∫_0 ^1 cos(max(x^3 ,y^(3/2) ))dxdy=A old Quation By mr,univers x^3 =t,y^(3/2) =s A=(2/9)∫_0 ^1 ∫_0 ^1 cos(max(t,s))t^(−(2/3)) s^(−(1/3)) dtds ∫_0 ^1 ∫_0 ^1 cos(max(t,s))t^(−(2/3)) s^(−(1/3)) dtds=∫_0 ^1 t^(−(2/3)) ∫_t ^1 cos(s)s^(−(1/3)) dsdt_(=C) +∫_0 ^1 s^(−(1/3)) ∫_s ^1 cos(t)t^(−(2/3)) dtds_(=B) c=∫_0 ^1 t^(−(2/3)) ∫_t ^1 Σ_(n≥0) (((−1)^n )/(2n!)).s^(2n−(1/3)) dsdt=Σ_(n≥0) (((−1)^n )/(2n!))∫_0 ^1 t^(−(2/3)) ∫_t ^1 s^(2n−(1/3)) dsdt =Σ_(n≥0) (((−1)^n )/((2n)!))∫_0 ^1 ((1−t^(2n+(2/3)) )/(2n+(2/3))).t^(−(2/3)) dt =Σ_(n≥0) (((−1)^n )/((2n)!(2n+(2/3)))).(3−(1/(2n+1)))=3Σ_(n≥0) (((−1)^n )/((2n+1)!))=3sin(1) B=∫_0 ^1 s^(−(1/3)) ∫_s ^1 Σ_(n≥0) (((−1)^n )/((2n)!))t^(2n−(2/3)) dtds =Σ(((−1)^n )/((2n)!))∫_0 ^1 s^(−(1/3)) .(((1−s^(2n+(1/3)) )/(2n+(1/3))))ds =Σ_(n≥0) (((−1)^n )/((2n)!(2n+(1/3))))((3/2)−(1/(2n+1))) =(3/2)Σ_(n≥0) (((−1)^n )/((2n+1)!))=(3/2)sin(1) A=(2/9)(c+B)=(2/9)((3/2)sin(1)+3sin(1)) =sin(1) ∫_0 ^1 ∫_0 ^1 cos(Max(x^3 ,y^(3/2) ))dxdy=sin(1)

$$\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{cos}\left(\mathrm{max}\left(\mathrm{x}^{\mathrm{3}} ,\mathrm{y}^{\frac{\mathrm{3}}{\mathrm{2}}} \right)\right)\mathrm{dxdy}=\mathrm{A} \\ $$$$\mathrm{old}\:\mathrm{Quation}\:\mathrm{By}\:\mathrm{mr},\mathrm{univers} \\ $$$$\mathrm{x}^{\mathrm{3}} =\mathrm{t},\mathrm{y}^{\frac{\mathrm{3}}{\mathrm{2}}} =\mathrm{s} \\ $$$$\mathrm{A}=\frac{\mathrm{2}}{\mathrm{9}}\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{cos}\left(\mathrm{max}\left(\mathrm{t},\mathrm{s}\right)\right)\mathrm{t}^{−\frac{\mathrm{2}}{\mathrm{3}}} \mathrm{s}^{−\frac{\mathrm{1}}{\mathrm{3}}} \mathrm{dtds} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{cos}\left(\mathrm{max}\left(\mathrm{t},\mathrm{s}\right)\right)\mathrm{t}^{−\frac{\mathrm{2}}{\mathrm{3}}} \mathrm{s}^{−\frac{\mathrm{1}}{\mathrm{3}}} \mathrm{dtds}=\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{t}^{−\frac{\mathrm{2}}{\mathrm{3}}} \int_{\mathrm{t}} ^{\mathrm{1}} \mathrm{cos}\left(\mathrm{s}\right)\mathrm{s}^{−\frac{\mathrm{1}}{\mathrm{3}}} \mathrm{dsdt}_{=\mathrm{C}} \\ $$$$+\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{s}^{−\frac{\mathrm{1}}{\mathrm{3}}} \int_{\mathrm{s}} ^{\mathrm{1}} \mathrm{cos}\left(\mathrm{t}\right)\mathrm{t}^{−\frac{\mathrm{2}}{\mathrm{3}}} \mathrm{dtds}_{=\mathrm{B}} \\ $$$$\mathrm{c}=\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{t}^{−\frac{\mathrm{2}}{\mathrm{3}}} \int_{\mathrm{t}} ^{\mathrm{1}} \underset{\mathrm{n}\geqslant\mathrm{0}} {\sum}\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\mathrm{2n}!}.\mathrm{s}^{\mathrm{2n}−\frac{\mathrm{1}}{\mathrm{3}}} \mathrm{dsdt}=\underset{\mathrm{n}\geqslant\mathrm{0}} {\sum}\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\mathrm{2n}!}\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{t}^{−\frac{\mathrm{2}}{\mathrm{3}}} \int_{\boldsymbol{\mathrm{t}}} ^{\mathrm{1}} \mathrm{s}^{\mathrm{2n}−\frac{\mathrm{1}}{\mathrm{3}}} \boldsymbol{\mathrm{dsdt}} \\ $$$$=\underset{\mathrm{n}\geqslant\mathrm{0}} {\sum}\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\left(\mathrm{2n}\right)!}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}−\mathrm{t}^{\mathrm{2n}+\frac{\mathrm{2}}{\mathrm{3}}} }{\mathrm{2n}+\frac{\mathrm{2}}{\mathrm{3}}}.\mathrm{t}^{−\frac{\mathrm{2}}{\mathrm{3}}} \mathrm{dt} \\ $$$$=\underset{\mathrm{n}\geqslant\mathrm{0}} {\sum}\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\left(\mathrm{2n}\right)!\left(\mathrm{2n}+\frac{\mathrm{2}}{\mathrm{3}}\right)}.\left(\mathrm{3}−\frac{\mathrm{1}}{\mathrm{2n}+\mathrm{1}}\right)=\mathrm{3}\underset{\mathrm{n}\geqslant\mathrm{0}} {\sum}\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\left(\mathrm{2n}+\mathrm{1}\right)!}=\mathrm{3sin}\left(\mathrm{1}\right) \\ $$$$\mathrm{B}=\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{s}^{−\frac{\mathrm{1}}{\mathrm{3}}} \int_{\mathrm{s}} ^{\mathrm{1}} \underset{\mathrm{n}\geqslant\mathrm{0}} {\sum}\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\left(\mathrm{2n}\right)!}\mathrm{t}^{\mathrm{2n}−\frac{\mathrm{2}}{\mathrm{3}}} \mathrm{dtds} \\ $$$$=\Sigma\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\left(\mathrm{2n}\right)!}\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{s}^{−\frac{\mathrm{1}}{\mathrm{3}}} .\left(\frac{\mathrm{1}−\mathrm{s}^{\mathrm{2n}+\frac{\mathrm{1}}{\mathrm{3}}} }{\mathrm{2n}+\frac{\mathrm{1}}{\mathrm{3}}}\right)\mathrm{ds} \\ $$$$=\underset{\mathrm{n}\geqslant\mathrm{0}} {\sum}\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\left(\mathrm{2n}\right)!\left(\mathrm{2n}+\frac{\mathrm{1}}{\mathrm{3}}\right)}\left(\frac{\mathrm{3}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{2n}+\mathrm{1}}\right) \\ $$$$=\frac{\mathrm{3}}{\mathrm{2}}\underset{\mathrm{n}\geqslant\mathrm{0}} {\sum}\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\left(\mathrm{2n}+\mathrm{1}\right)!}=\frac{\mathrm{3}}{\mathrm{2}}\mathrm{sin}\left(\mathrm{1}\right) \\ $$$$\mathrm{A}=\frac{\mathrm{2}}{\mathrm{9}}\left(\mathrm{c}+\mathrm{B}\right)=\frac{\mathrm{2}}{\mathrm{9}}\left(\frac{\mathrm{3}}{\mathrm{2}}\mathrm{sin}\left(\mathrm{1}\right)+\mathrm{3sin}\left(\mathrm{1}\right)\right) \\ $$$$=\mathrm{sin}\left(\mathrm{1}\right) \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{cos}\left(\mathrm{Max}\left(\mathrm{x}^{\mathrm{3}} ,\mathrm{y}^{\frac{\mathrm{3}}{\mathrm{2}}} \right)\right)\mathrm{dxdy}=\mathrm{sin}\left(\mathrm{1}\right) \\ $$$$ \\ $$

Commented by universe last updated on 06/Nov/23

thanks sir

$${thanks}\:{sir} \\ $$

Commented by witcher3 last updated on 06/Nov/23

withe Pleasur

$$\mathrm{withe}\:\mathrm{Pleasur} \\ $$

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