Let $f(x)>0$ for all $x \in \mathbb{R}$ and $f(x)$ is bounded. If $\mathop {\lim }\limits_{n \to \infty } \sum_{r-1}^n a^{r-1} \int_{(r-1) a}^{r a} \frac{f(x) d x}{f(x)+f(2 r a-a-x)}=\frac{3}{5}$ where $0< a< 1$, then the value(s) of a is are

If $f(x)=\int\limits_0^{\sin ^2 x} \sin ^{-1} \sqrt{t} d t$ and $g(x)=\int\limits_0^{\cos ^2 x} \cos ^{-1} \sqrt{t} d t$, then the value of $f(x)+g(x)$ is

The value of $\int\limits_{-100}^{100} \frac{\left(x+x^3+x^5\right)}{\left(1+x^2+x^4+x^6\right)} d x$ is

$$ \text { The points of extremum of } \int_\limits0^{x^2} \frac{t^2-5 t+4}{2+e^t} d t \text { are } $$