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Question Number 143191 by mohammad17 last updated on 11/Jun/21

Answered by mathmax by abdo last updated on 11/Jun/21

$${\int }_0^3{\int }_{\sqrt{\frac{\mathrm{x}}{3}}}^1{({\mathrm{y}}^3+1)}^3\mathrm{dydx}={\int }_0^3\left({\int }_{\sqrt{\frac{\mathrm{x}}{3}}}^1({\mathrm{y}}^9+{3\mathrm{y}}^6+{3\mathrm{y}}^3+1)\mathrm{dy}\right)\mathrm{dx}$$$$={\int }_0^3{[\frac{{\mathrm{y}}^{10}}{10}+\frac{3}{7}{\mathrm{y}}^7+\frac{3}{4}{\mathrm{y}}^4+\mathrm{y}]}_{\sqrt{\frac{\mathrm{x}}{3}}}^1\mathrm{dx}$$$$={\int }_0^3(\frac{1}{10}+\frac{3}{7}+\frac{3}{4}+1-\frac{1}{10}{\left(\frac{\mathrm{x}}{3}\right)}^5-\frac{3}{7}{\left(\frac{\mathrm{x}}{3}\right)}^{\frac{7}{2}}-\frac{3}{4}{\left(\frac{\mathrm{x}}{3}\right)}^2-\sqrt{\frac{\mathrm{x}}{3}})\mathrm{dx}$$$$=3(\frac{1}{10}+\frac{3}{7}+\frac{3}{4}+1)-\frac{1}{10.3^5}{\int }_0^3{\mathrm{x}}^5\mathrm{dx}-\frac{3}{7.3^{\frac{7}{2}}}{\int }_0^3{\mathrm{x}}^{\frac{7}{2}}\mathrm{dx}$$$$-\frac{3}{4.3^2}{\int }_0^3{\mathrm{x}}^2\mathrm{dx}-\frac{1}{3^{\frac{1}{2}}}{\int }_0^3{\mathrm{x}}^{\frac{1}{2}}\mathrm{dx}$$$$=\frac{3}{10}+\frac{9}{7}+\frac{9}{4}+3-\frac{1}{10.3^5.6}{3}^6-\frac{3}{7.3^{\frac{7}{2}}.\frac{9}{2}}{3}^{\frac{9}{2}}$$$$-\frac{3}{4.3^2.3}{3}^2-\frac{1}{3^{\frac{1}{2}}.\frac{3}{2}}{3}^{\frac{3}{2}}=....$$

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