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Question Number 27792 by abdo imad last updated on 14/Jan/18

$$find{lim}_{x->1}{\int }_x^{x^2}\frac{dt}{ln\left(t\right)}.$$

Commented byabdo imad last updated on 15/Jan/18

$$iffcontinuein[a,b]andgintegrablein[a,b]\mathrm{\exists }c\in ]a,b[/$$ $${\int }_a^{b}f\left(x\right)g\left(x\right)dx=f\left(c\right){\int }_a^{b}g\left(x\right)dxso\mathrm{\exists }{c}_x\in ]x,x^2[/$$ $${\int }_x^{x^2}\frac{dt}{lnt}=\frac{1}{lnc}{\int }_x^{x^2}dx=\frac{x^2-x}{{lnc}_x}=\frac{x(x-1)}{ln\left(c_x\right)}butx->1\iff c_x->1$$ $$so{lim}_{x->1}{\int }_x^{x^2}\frac{dt}{lnt}={lim}_{c_{x->1}}\frac{c_x(c_x^{}-1)}{{lnc}_x}$$ $${lim}_{c_{x->1}}\frac{1}{\frac{{lnc}_x}{c_x-1}}=1.(wecanconfusexand{c}_{x}brcause$$ $$x\in V\left(1\right)and{c}_x\in V\left(1\right).$$

Commented byprakash jain last updated on 16/Jan/18

$$\mathrm{understood}\mathrm{rest}\mathrm{of}\mathrm{the}\mathrm{solution}\mathrm{but}$$ $$\mathrm{What}\mathrm{is}V\left(1\right)?\mathrm{is}\mathrm{it}\mathrm{vicinity}\mathrm{of}1?$$ $$\mathrm{PS}:\mathrm{there}\mathrm{are}\mathrm{symbols}\mathrm{in}\mathrm{the}\mathrm{app}\mathrm{like}$$ $$\to ,\mathrm{\infty }\mathrm{which}\mathrm{you}\mathrm{can}\mathrm{use}\mathrm{press}$$ $$\mathrm{green}\int \mathrm{button}\mathrm{on}\mathrm{bottom}\mathrm{left}\mathrm{corner}.$$

Commented byabdo imad last updated on 16/Jan/18

$$x\in V\left(1\right)meanthatxissonearfrom$$ $$1.$$

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