JEE Advanced 2016 · Paper 1, Mathematics question 52

Function. Let $F(x)=\displaystyle\int_0^x\frac{t^2}{1+t^4}\,dt-2x+1$.

End values.

• $F(0)=1>0$.
• $F(1)<\frac12-1<0$, since the integrand is below $\frac12$ on $[0,1]$.

Monotonic. $F'(x)=\dfrac{x^2}{1+x^4}-2<0$, so $F$ is strictly decreasing and crosses zero exactly once.

Answer: 1