Given `int_(1)^(2) e^(x^(2))dx=a`, the value of `int_(e )^(e^(4)) sqrt(log_(e )x)dx`, is

Options

A. `e^(4)-e`
B. `e^(4)-a`
C. `2e^(4)-a`
D. `2e^(4)-e-a`

Answer

D

Explanation

We have,
`I=overset(e^(4))underset(e )int sqrt(log_(e )x)dx=overset(2)underset(1)int sqrt(t^(2))e^(t^(2))2t" dt", "where" log_(e )x=t^(2)`
`rArrI=2overset(2)underset(1)int t^(2)e^(t^(2))2t" dt"`
`rArr I=overset(2)underset(1)int underset(I)(t)e^(t^(2))underset(II)(2t)dt=[te^(t^(2))]_(1)^(2)-overset(2)underset(1)int e^(t^(2))dt=2e^(4)-e-a`