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Evaluate int(0)^(1) (e^x(2-x^2)dx)/((1-x...

Evaluate `int_(0)^(1) (e^x(2-x^2)dx)/((1-x)sqrt(1-x^2))`

A

`(sqrt(e))/2(sqrt(3)+1)`

B

`(sqrt(3e))/2`

C

`2e-1`

D

`sqrt(e/3)`

Text Solution

Verified by Experts

The correct Answer is:
C
`int_(-1)^(1//2)(e^(x)(2-x^(2))dx)/((1-x)sqrt(1-x^(2)))=int_(-1)^(1//2)(e^(x)(1-x^(2)+1))/((1-x)sqrt(1-x^(2)))`
`=int_(-1)^(1//2) e^(x)[(sqrt(1+x))/(1-x)+1/((1-x)sqrt(1-x^(2)))]dx`
`=e^(x)sqrt(1+x)/(1-x)|_(-1)^(1//2)=sqrt(3e)`
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