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Question Number 97439 by bemath last updated on 08/Jun/20

$$\underset{\frac{\sqrt{2}}{2}}{\stackrel{1}{\int }}(\frac{{\mathrm{x}}^3}{2}+\frac{1}{6\mathrm{x}})\sqrt{1+{(\frac{{3\mathrm{x}}^2}{2}-\frac{1}{{6\mathrm{x}}^2})}^2}\mathrm{dx}$$

Commented by john santu last updated on 08/Jun/20

$$\left(1\right)1+{(\frac{{3\mathrm{x}}^2}{2}-\frac{1}{{6\mathrm{x}}^2})}^2=1+\frac{{9\mathrm{x}}^4}{4}-\frac{1}{2}+\frac{1}{{36\mathrm{x}}^4}$$$$=\frac{{9\mathrm{x}}^4}{4}+\frac{1}{2}+\frac{1}{{36\mathrm{x}}^4}={(\frac{{3\mathrm{x}}^2}{2}+\frac{1}{{6\mathrm{x}}^2})}^2$$$$\left(2\right)\sqrt{1+{(\frac{{3\mathrm{x}}^2}{2}-\frac{1}{{6\mathrm{x}}^2})}^2}=\sqrt{{(\frac{{3\mathrm{x}}^2}{2}+\frac{1}{{6\mathrm{x}}^2})}^2}$$$$=\frac{{3\mathrm{x}}^2}{2}+\frac{1}{{6\mathrm{x}}^2}$$$$\left(3\right)\underset{\frac{\sqrt{2}}{2}}{\stackrel{1}{\int }}(\frac{{\mathrm{x}}^3}{2}+\frac{1}{6\mathrm{x}})(\frac{{3\mathrm{x}}^2}{2}+\frac{1}{{6\mathrm{x}}^2})\mathrm{dx}=$$$$\underset{\frac{\sqrt{2}}{2}}{\stackrel{1}{\int }}(\frac{{3\mathrm{x}}^5}{4}+\frac{\mathrm{x}}{3}+\frac{1}{{36\mathrm{x}}^3})\mathrm{dx}=$$$${[\frac{{\mathrm{x}}^6}{8}+\frac{{\mathrm{x}}^2}{6}-\frac{1}{{72\mathrm{x}}^2}]}_{\frac{\sqrt{2}}{2}}^1$$$$=\frac{1}{8}(1-\frac{1}{8})+\frac{1}{6}(1-\frac{1}{2})-\frac{1}{72}(1-2)$$$$=\frac{7}{64}+\frac{1}{12}+\frac{1}{72}=\frac{7}{64}+\frac{7}{72}$$$$=\frac{63+56}{576}=\frac{119}{576}\therefore$$

Commented by bemath last updated on 08/Jun/20

$$\mathrm{thank}\mathrm{you}$$

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