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Question Number 124853 by bemath last updated on 06/Dec/20

$${\int }_1^{x^3+5x}f\left(t\right)dt=2x$$$$thenf\left(18\right)=?$$

Answered by liberty last updated on 06/Dec/20

$$\frac{d}{dx}\left[{\int }_1^{x^3+5x}f\left(t\right)dt\right]=\frac{d}{dx}\left(2x\right)$$$$(3x^2+5)f(x^3+5x)=2\Rightarrow f(x^3+5x)=\frac{2}{3x^2+5}$$$$put{x}^3+5x=18;(x-2)(x^2+2x+9)=0$$$$\Rightarrow x_1=2\wedge f\left(18\right)=\frac{2}{3\times 4+5}=\frac{2}{17}$$$$\Rightarrow x_{2,3}=\frac{-2\pm 4i\sqrt{2}}{2}=-1\pm 2i\sqrt{2}$$$$wegetf\left(18\right)=\frac{2}{3(-7-4i\sqrt{2})+5}=\frac{2}{-16-12i\sqrt{2}}=\frac{1}{-8-6i\sqrt{2}}$$$$andf\left(18\right)=\frac{2}{3(-7+4i\sqrt{2})+5}=\frac{2}{-16+12i\sqrt{2}}=\frac{1}{-8+6i\sqrt{2}}$$

Answered by mathmax by abdo last updated on 06/Dec/20

$$\mathrm{by}\mathrm{derivation}\mathrm{we}\mathrm{get}({3\mathrm{x}}^2+5)\mathrm{f}({\mathrm{x}}^3+5\mathrm{x})=2$$$$\mathrm{x}=2\Rightarrow (12+5)\mathrm{f}(8+10)=2\Rightarrow 17\mathrm{f}\left(18\right)=2\Rightarrow \mathrm{f}\left(18\right)=\frac{2}{17}$$

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