Let `I_(1)=int_(0)^(1) (e^(x))/(1+x)dx` and `I_(2)=int_(0)^(1) (x^(2))/(e^(x^(3))(2-x^(3)))dx`.
Then, `(I_(1))/(I_(2))` is equal to

Options

A. `(3)/(e )`
B. `(e )/(3)`
C. 3e
D. `(1)/(3e)`

Answer

C

Explanation

We have,
`I_(1)=overset(1)underset(0)int (e^(x))/(1+x)dx`and `I_(2)=overset(1)underset(0)int (x^(2))/(e^(x^(3))(2x^(3)))dx`
`:. I_(2)=(1)/(3)overset(1)underset(0)int (1)/(e^(x^(3))(2-x^(3)))3x^(2)dx`
`rArr I_(2)=(1)/(3)overset(1)underset(0)int (1)/(e(t)(2-t))dt`where `t=x^(3)`
`rArr I_(2)=(1)/(3)overset(1)underset(0)int (1)/(e^(1-t)(1+t))dt" "[:' overset(a)underset(0)int f(x)dx=overset(a)underset(0)int f(a-x)]`
`rArr I_(2)=(1)/(3e)overset(1)underset(0)int (e^(t))/(1+t)dt rArr I_(2)=(1)/(3e)I_(2)=(1)/(3e)I_(1)rArr(I_(1))/(I_(2))=3e`