If ∫$$dx1+x21$$−$$x2=F(x)$$ and $$F(1)=0then$$ for x $$0$$, F (x) is equal to

Question

If ∫$$dx1+x21$$−$$x2=F(x)$$ and  $$F(1)=0then$$ for x > $$0$$, F (x) is equal to

Infinity Learn · Accepted Answer

Putting $$x=1t$$ so that $$dx=$$−$$1t2dtWe$$ have                    ∫$$dx1+x21$$−$$x2=$$∫−$$tdt1+t2t2$$−$$1=$$−∫$$dx2+u2t2$$−$$1=u2=$$−$$12tan$$−$$1$$⁡$$u2+C=$$−$$12tan$$−$$1$$⁡$$121$$−$$x2x+CSince$$ $$F(1)$$ $$=$$ $$0$$, so C $$=$$ $$0$$. $$HenceF(x)=$$−$$12$$$$π$$$$2$$−$$cot$$−$$1$$⁡$$121$$−$$x2x$$                                                                 $$tan$$−$$1$$⁡θ$$+$$$$cot$$−$$1$$⁡θ$$=$$π$$/2=$$−π$$22+12tan$$−$$1$$⁡$$2x1$$−$$x2cot$$−$$1$$⁡θ$$=$$$$tan$$−$$1$$⁡$$1$$θ, for θ>$$0$$ and $$cot$$−$$1$$⁡θ$$=$$$$tan$$−$$1$$⁡$$1$$θ$$+$$π, for θ<$$0$$

If $\int \frac{d x}{(1 + x^{2}) \sqrt{1 - x^{2}}} = F (x)$ and $F (1) = 0$
then for $x > 0, F (x$ ) is equal to

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If ∫dx1+x21−x2=F(x) and F(1)=0then for x > 0, F (x) is equal to

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Ready to Test Your Skills?

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If $\int \frac{d x}{(1 + x^{2}) \sqrt{1 - x^{2}}} = F (x)$ and $F (1) = 0$
then for $x > 0, F (x$ ) is equal to