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Question Number 157655 by tounghoungko last updated on 26/Oct/21

  x^2  f(x^3 )+(1/((1+x)^2 )) f(((1−x)/(1+x)))=4x^3 (1+x^4 )^5    ∫_( 0) ^( 1) f(x) dx =?

$$\:\:{x}^{\mathrm{2}} \:{f}\left({x}^{\mathrm{3}} \right)+\frac{\mathrm{1}}{\left(\mathrm{1}+{x}\right)^{\mathrm{2}} }\:{f}\left(\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}}\right)=\mathrm{4}{x}^{\mathrm{3}} \left(\mathrm{1}+{x}^{\mathrm{4}} \right)^{\mathrm{5}} \\ $$$$\:\int_{\:\mathrm{0}} ^{\:\mathrm{1}} {f}\left({x}\right)\:{dx}\:=? \\ $$

Commented by cortano last updated on 26/Oct/21

= ((65)/3)

$$=\:\frac{\mathrm{65}}{\mathrm{3}} \\ $$

Answered by mindispower last updated on 26/Oct/21

∫_0 ^1 x^2 f(x^3 )dx=(1/3)∫_0 ^1 f(x)dx  ∫_0 ^1 f(x)dx=2∫_0 ^1 ((f(((1−x)/(1+x))))/((1+x)^2 ))dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{\mathrm{2}} {f}\left({x}^{\mathrm{3}} \right){dx}=\frac{\mathrm{1}}{\mathrm{3}}\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx}=\mathrm{2}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{f}\left(\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}}\right)}{\left(\mathrm{1}+{x}\right)^{\mathrm{2}} }{dx} \\ $$

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