`L e tI_1=int_0^1(e^xdx)/(1+x) a n dI_2=int_0^1(x^2dx)/(e^(x^3)(2-x^3))dot tehn (I_1)/(I_2) is equal to` (a)3/e` (b) `e/3` (c) `3e` (d) `1/(3e)`

Options

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

Answer

C

Explanation

`I_(1)=int_(0)^(1)(e^(x)dx)/(1+x), I_(2)=int_(0)^(1)(x^(2)dx)/(e^(x^(2))(2-x^(3)))`
In `I_(2)` put `1-x^(3)=t`
`:. I_(2)=1/3 int_(1)^(0) (-dt)/(e^(1-t)(1+t))=1/(3e)int_(0)^(1)(e^(t)dt)/(1+t)=1/(3e)I_(1)`
`:. (I_(1))/(I_(2))=3e`