For what values of $m$ do the line $$y =mx$$ and the curve $$y=\frac x{1+x^2}$$ enclose a region? Find the area of the region.
I did the integration but I was unable to solve this question. Please help me in this question ?
2026-04-13 01:38:15.1776044295
Find value of $m$ such that $y =mx$ and $y=\frac x{x^2+1}$ enclose a region
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Solve $y=mx=\frac x{1+x^2}$ to get $x=0,\pm \sqrt{\frac{1-m}m}$. Thus, for $0<m<1$, the intercept value of $x$ is real and two curves enclose a region. Its area is integrated as $$ \int_0^{ \sqrt{\frac{1-m}m}}\left(\frac x{1+x^2}-mx\right) dx=\frac12\ln\frac1m-\frac12(1-m) $$