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

If f(x)= x^3 −((3x^2 )/2) +x + (1/4). Then ∫_(1/4) ^(3/4) f(f(x))dx =?

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

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

Hint given: Replace x by 1−x in function and then arrange to get result.

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

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

Thanks sir.

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

Commented by MJS last updated on 01/Sep/18

f(1−x)=−x^3 +(3/2)x^2 −x+(3/4)=1−f(x) f(f(1−x))=1−f(f(x)) ∫_(1/4) ^(3/4) f(f(1−x))dx=∫_(1/4) ^(3/4) 1dx−∫_(1/4) ^(3/4) f(f(x))dx but ∫_(1/4) ^(3/4) f(f(1−x))dx∫_(3/4) ^(1/4) f(f(x))dx=−∫_(1/4) ^(3/4) f(f(x))dx ⇒ 2∫_(1/4) ^(3/4) f(f(x))dx=∫_(1/4) ^(3/4) 1dx ⇒ ∫_(1/4) ^(3/4) f(f(x))dx=(1/2)∫_(1/4) ^(3/4) dx=(1/4)

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

Answered by Meritguide1234 last updated on 03/Sep/18

Commented by Meritguide1234 last updated on 03/Sep/18

i use word to type maths. i live in taiwan.

$${i}\:{use}\:{word}\:{to}\:{type}\:{maths}. \\ $$$${i}\:{live}\:{in}\:{taiwan}. \\ $$

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