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Question Number 167520 by MikeH last updated on 18/Mar/22

Given that f(x) = ∫_x ^(2x) (1/( (√(1+t^4 ))))dt  (a) state its domain  (b) is f(x) even or odd?

$$\mathrm{Given}\:\mathrm{that}\:{f}\left({x}\right)\:=\:\int_{{x}} ^{\mathrm{2}{x}} \frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{t}^{\mathrm{4}} }}{dt} \\ $$$$\left(\mathrm{a}\right)\:\mathrm{state}\:\mathrm{its}\:\mathrm{domain} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{is}\:{f}\left({x}\right)\:\mathrm{even}\:\mathrm{or}\:\mathrm{odd}? \\ $$

Answered by aleks041103 last updated on 18/Mar/22

(a) obv. f(x) is defined for ∀x∈R.  (b) f(−x)=∫_(−x) ^(−2x) (dt/( (√(1+t^4 ))))=  =−∫_(−x) ^(−2x) ((d(−t))/( (√(1+(−t)^4 ))))=−∫_x ^( 2x) (dt/( (√(1+t^4 ))))=−f(x)  ⇒f(−x)=−f(x)  ⇒f(x) is odd.

$$\left({a}\right)\:{obv}.\:{f}\left({x}\right)\:{is}\:{defined}\:{for}\:\forall{x}\in\mathbb{R}. \\ $$$$\left({b}\right)\:{f}\left(−{x}\right)=\int_{−{x}} ^{−\mathrm{2}{x}} \frac{{dt}}{\:\sqrt{\mathrm{1}+{t}^{\mathrm{4}} }}= \\ $$$$=−\int_{−{x}} ^{−\mathrm{2}{x}} \frac{{d}\left(−{t}\right)}{\:\sqrt{\mathrm{1}+\left(−{t}\right)^{\mathrm{4}} }}=−\int_{{x}} ^{\:\mathrm{2}{x}} \frac{{dt}}{\:\sqrt{\mathrm{1}+{t}^{\mathrm{4}} }}=−{f}\left({x}\right) \\ $$$$\Rightarrow{f}\left(−{x}\right)=−{f}\left({x}\right) \\ $$$$\Rightarrow{f}\left({x}\right)\:{is}\:{odd}. \\ $$

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