`int_(0)^(pi) e^(x) sin x dx`, किसके बराबर है ?

Options

A. `(e^(pi)+1)/(2)`
B. `(e^(pi)-1)/(2)`
C. `e^(pi)+1`
D. `(e^(pi)+1)/(4)`

Answer

A

Explanation

माना `l=int_(0)^(pi)sin x dx`
`=[sinx.e^(x)]_(0)^(pi)-int_(0)^(pi)[(d)/(dx){sin x}.e^(x)]dx`
`=[sinx.e^(x)]_(0)^(pi)-int_(0)^(pi)cos x.e^(x)dx`
`0-{[cos x.e^(x)]_(0)^(pi)+int_(0)^(pi)sin x.e^(x)dx}`
`rArr l=-[-e^(pi)-1]-l`
`rArr l+l=e^(pi)+1`
`rArr2l=e^(pi)+1`
`rArr l=(e^(pi)+1)/(2)`