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If lamda=int(0)^(1)(e^(t))/(1+t), then f...

If `lamda=int_(0)^(1)(e^(t))/(1+t)`, then find the value of `int_(0)^(1)e^(t)log_(e)(1+t)dt`

A

k

B

2k

C

e In 2-k

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
C
`k=underset(0)overset(1)int (e^(t))/(1+t)dt=underset(0)overset(1)int e^(t)(1)/(underset(I" "II" ")(1+t))dt`
`rArr k=[e^(t)log(1+t)]_(0)^(1)-underset(0)overset(1)int underset(II)e^(t) log(underset(I" ")(1+1))dt`
`rArr k=e log_(e )2-underset(0)overset(1)int e^(t) log(1+t)dt`
`rArr underset(0)overset(1)int e^(t) log(1+t)dt=e" In "2-k`
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