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The value of the definite integral int0^...

The value of the definite integral `int_0^1(1+e^-x^2)dx` (b) 2 `1+e^(-1)` (d) none of these

A

`-1`

B

2

C

`1+e^(-1)`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
D
If f (x) is a continuous function defined on [ a , b] , then `m ( b - a)le int_(a)^(b)f (x) dx le M ( b - a)`
where , M and m are maximum and minimum values respectively of f (x) in [ a , b ] .
Here , f(x) `= 1 + e ^(-x^(2))` is continuous in [ 0 , 1].
Now , `0lt x lt 1 rArrx^(2)lt x rArr e^(x^(2))lt e^(x) rArr e^(-x^(2))gte^(-x)`
Again , `0 lt x lt 1 rArr x^(2) gt 0 rArr e^(x^(2))gt e^(0) rArr e^(-x^(2))lt1`
`:. e^(-x)lt e^(-x^(2))lt1 , AA x in [ 0 , 1 ]`
`rArr 1 + e^(-x) lt 1 +e^(-x^(2))lt 2 , AA x in [ 0 , 1 ]`
`rArr int_(0)^(1) ( 1+e^(-x)) dx lt int_(0)^(1)(1+e^(x^(2))) dx lt int_(0)^(1) 2 dx`
`rArr 2 - (1)/(e) lt int_(0)^(1) ( 1 + e^(-x^(2))) dx lt 2`
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