Suppose and are 2 continuous functions defined for .
and
. The value of is
1)
2)
3)
4)
. The value of - is
5)
6)
7)
8)
. The value of is
9)
10)
11)
12)
Details for entering the answer:
If your answer comes as option 1 for A, option 7 for B and option 12 for C, then write your answer as 1712.
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Let's observe f ( x ) . f ( x ) = ∫ 0 1 e x + t . f ( t ) d t f ( x ) = e x ∫ 0 1 e t . f ( t ) d t Let's define a number α as α = ∫ 0 1 e t . f ( t ) d t Hence we can write that:- f ( x ) = α e x Hence we can write that α = ∫ 0 1 e t . α e t d t We will be urged to cancel α and get a false expression.
It means only one thing that it will be wrong to cancel α because α = 0 .
Hence we get f ( x ) = 0 .
Now Let's observe g ( x ) . g ( x ) = e x ∫ 0 1 e t . g ( t ) d t + x Again let us define β as follows:- β = ∫ 0 1 e t . g ( t ) d t Hence we can write :- g ( x ) = β e x + x Again we can write that:- β = ∫ 0 1 e t ( β e t + t ) d t β = β ∫ 0 1 e 2 t d t + ∫ 0 1 e t t d t β = β ( 2 e 2 − 1 ) + 1 β = 3 − e 2 2 Hence we get that:- g ( x ) = 3 − e 2 2 e x + x Now we can get our desired answers.