I'm having a lot of trouble finding the inverse of this function, I'm not quite sure even where to begin with..
$$f(x) = x\sqrt{1-x^2} + 2\int_0^1 \sqrt{1-t^2} \, dt$$
Dom is $\mathbb R$.
Could you guys offer help?
2026-04-02 12:59:12.1775134752
About the inverse of a function $f$ with integral
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The integral is a constant, so I will denote it with $C$. You have $$y=x\sqrt{1-x^2}+C$$ $$y-C=x\sqrt{1-x^2}$$ $$(y-C)^2=x^2*(1-x^2)$$ $$(y-C)^2=x^2-x^4$$ Substitue $u=x^2$. $$u^2-u+(y-C)^2=0$$
From there, use the quadratic formula to get your answer, and then fix the domain and range so that everything is real, unless you want it to be complex.