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Question Number 138205 by Bekzod Jumayev last updated on 11/Apr/21

Commented by Jamshidbek last updated on 11/Apr/21

$${\mathrm{Og}}^{\prime }\mathrm{ayni}\mathrm{kun}\mathrm{savolini}\mathrm{oziz}\mathrm{yechmasizmi}?$$$$\mathrm{Yoki}\mathrm{kechqurun}\mathrm{kanalga}\mathrm{tashlanadiku}\mathrm{yechim}.{\mathrm{O}}^{\prime }\mathrm{shanda}$$$${\mathrm{o}}^{\prime }\mathrm{rganib}\mathrm{olardizda}$$

Commented by Bekzod Jumayev last updated on 13/Apr/21

$$Uzuryechimni{o}^{\prime }rganmoqchiedim$$$$kanalgajoylashniyatimyoqedi$$$$$$

Commented by Jamshidbek last updated on 13/Apr/21

$$\mathrm{Uzr}\mathrm{men}{\mathrm{noto}}^{\prime }{\mathrm{g}}^{\prime }\mathrm{ri}\mathrm{tushunib}\mathrm{olibman}.\mathrm{Men}$$$$\mathrm{sizni}\mathrm{kanalga}\mathrm{joylash}\mathrm{uchun}{\mathrm{so}}^{\prime }\mathrm{rayapti}$$$$\mathrm{deb}{\mathrm{o}}^{\prime }\mathrm{ylabman}.\mathrm{Uzr}\mathrm{ming}\mathrm{bor}$$

Answered by phanphuoc last updated on 11/Apr/21

$$x=costorx=sint$$$$$$

Answered by MJS_new last updated on 11/Apr/21

$$\int \frac{x^2}{x+\sqrt{1-x^2}}dx=$$$$[t=\frac{x}{\sqrt{1-x^2}}\to dx={(1-x^2)}^{3/2}dt]$$$$=\int \frac{t^2}{(t+1){(t^2+1)}^2}dt=$$$$\left[{\mathrm{Ostrogradski}}^{\prime }\mathrm{s}\mathrm{Method}\right]$$$$=-\frac{t+1}{4(t^2+1)}-\frac{1}{4}\int \frac{t-1}{(t+1)(t^2+1)}dt=$$$$=-\frac{t+1}{4(t^2+1)}+\frac{1}{4}\int \frac{1}{t+1}-\frac{t}{t^2+1}dt=$$$$=-\frac{t+1}{4(t^2+1)}+\frac{1}{4}\ln (t+1)-\frac{1}{8}\ln (t^2+1)=$$$$=-\frac{t+1}{4(t^2+1)}+\frac{1}{8}\ln \frac{{(t+1)}^2}{t^2+1}=$$$$=\frac{x^2-1-x\sqrt{1-x^2}}{4}+\frac{1}{4}\ln (x+\sqrt{1-x^2})+C$$$$\Rightarrow \mathrm{answer}\mathrm{is}\frac{1}{4}$$

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