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Question Number 42709 by rahul 19 last updated on 01/Sep/18

$$\mathrm{If}\mathrm{f}\left(x\right)=x^3-\frac{3x^2}{2}+x+\frac{1}{4}.$$$$T\mathrm{hen}{\int }_{\frac{1}{4}}^{\frac{3}{4}}\mathrm{f}\left(\mathrm{f}\left(x\right)\right)\mathrm{d}x=?$$

Commented by rahul 19 last updated on 01/Sep/18

$$\mathrm{Hint}\mathrm{given}:\mathrm{Replace}x\mathrm{by}1-x\mathrm{in}$$$$\mathrm{function}\mathrm{and}\mathrm{then}\mathrm{arrange}\mathrm{to}\mathrm{get}\mathrm{result}.$$

Commented by rahul 19 last updated on 02/Sep/18

$$\mathrm{Thanks}\mathrm{sir}.$$

Commented by MJS last updated on 01/Sep/18

$$f(1-x)=-x^3+\frac{3}{2}{x}^2-x+\frac{3}{4}=1-f\left(x\right)$$$$f\left(f(1-x)\right)=1-f\left(f\left(x\right)\right)$$$$\underset{\frac{1}{4}}{\stackrel{\frac{3}{4}}{\int }}f\left(f(1-x)\right)dx=\underset{\frac{1}{4}}{\stackrel{\frac{3}{4}}{\int }}1dx-\underset{\frac{1}{4}}{\stackrel{\frac{3}{4}}{\int }}f\left(f\left(x\right)\right)dx$$$$\mathrm{but}\underset{\frac{1}{4}}{\stackrel{\frac{3}{4}}{\int }}f\left(f(1-x)\right)dx\underset{\frac{3}{4}}{\stackrel{\frac{1}{4}}{\int }}f\left(f\left(x\right)\right)dx=-\underset{\frac{1}{4}}{\stackrel{\frac{3}{4}}{\int }}f\left(f\left(x\right)\right)dx$$$$\Rightarrow 2\underset{\frac{1}{4}}{\stackrel{\frac{3}{4}}{\int }}f\left(f\left(x\right)\right)dx=\underset{\frac{1}{4}}{\stackrel{\frac{3}{4}}{\int }}1dx\Rightarrow \underset{\frac{1}{4}}{\stackrel{\frac{3}{4}}{\int }}f\left(f\left(x\right)\right)dx=\frac{1}{2}\underset{\frac{1}{4}}{\stackrel{\frac{3}{4}}{\int }}dx=\frac{1}{4}$$

Answered by Meritguide1234 last updated on 03/Sep/18

Commented by Meritguide1234 last updated on 03/Sep/18

$$iusewordtotypemaths.$$$$iliveintaiwan.$$

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