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

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

Answered by aleks041103 last updated on 18/Mar/22

$$\left(a\right)obv.f\left(x\right)isdefinedfor\mathrm{\forall }x\in \mathbb{R}.$$$$\left(b\right)f(-x)={\int }_{-x}^{-2x}\frac{dt}{\sqrt{1+t^4}}=$$$$=-{\int }_{-x}^{-2x}\frac{d(-t)}{\sqrt{1+{(-t)}^4}}=-{\int }_x^{2x}\frac{dt}{\sqrt{1+t^4}}=-f\left(x\right)$$$$\Rightarrow f(-x)=-f\left(x\right)$$$$\Rightarrow f\left(x\right)isodd.$$

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