If `f(x)=int_(x^(2))^(x^(2)+1)e^(-t^(2))dt` , then find the interval in which `f(x)` increases.

Explanation

We have `f(x)=int_(x^(2))^(x^(2)+1)e^(-t^(2))dt`
Differentiating w.r.t `x` we get
`impliesf'(x)=e^(-(x^(2)+1)^(2)).2x-e^(-x^(4).2x`
`=2x[e^(-(x^(2)+1)^(2))-e^(-x^(4))]`
Now `(x^(2)+1)^(2)gtx^(4)`
`:. -(x^(2)+1)^(2)lt -x^(4)`
`impliese^(-(x^(2)+1)^(2))lte^(-x^(4))`
`impliese^(-(x^(2)+1)^(2))-e^(-x^(4)lt0`
`:. f'(x)ge0,AA xle0`,
`:. f(x)` increases when `xle0`