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Question Number 124099 by aurpeyz last updated on 30/Nov/20

Answered by liberty last updated on 30/Nov/20

$$\Rightarrow 5\underset{0}{\stackrel{a}{\int }}(x^4-x^2)dx=3\underset{0}{\stackrel{a}{\int }}(x^2-x^3)dx$$$$\Rightarrow 5(\frac{a^5}{5}-\frac{a^3}{3})=3(\frac{a^3}{3}-\frac{a^4}{4})$$$$\Rightarrow a^5-\frac{5}{3}{a}^3-\frac{3a^3}{3}+\frac{3a^4}{4}=0$$$$\Rightarrow a^5+\frac{3a^4}{4}-\frac{8a^3}{3}=0;a^3(a^2+\frac{3a}{4}-\frac{8}{3})=0$$$$\Rightarrow {(a+\frac{3}{8})}^2-\frac{9}{64}-\frac{8}{3}=0$$$$\Rightarrow {(a+\frac{3}{8})}^2=\frac{27+64\times 8}{64\times 3}=\frac{539}{64\times 3}$$$$\Rightarrow a=-\frac{3}{8}\pm \frac{\sqrt{539}}{8\sqrt{3}}$$$$a_1=\frac{-3\sqrt{3}+\sqrt{539}}{8\sqrt{3}}=1.300497\approx 1.3$$$$$$

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