`int_(pi//2)^(pi) e^(x) ((1-sinx)/(1-cos x))dx` का मान ज्ञात कीजिएः

Explanation

`I=underset(pi//2)overset(pi)int e^(x) ((1-sinx)/(1-cos x))dx`
`=underset(pi//2)overset(pi)int ((1-2sin""x/2cos""x/2)/(2 sin^(2)""x/2))dx`
`=underset(pi//2)overset(pi)int ((1)/(2sin^(2)""x/2)-(cos ""pi/2)/(sin""pi/2))dx`
`=underset(pi//2)overset(pi)int (1/2"cosec"^2x-cot""x/2)dx`
`=underset(pi//2)overset(pi)int (-cot""x/2+1/2"cosec"^(2)x/2)dx` `=[e^(x)(-cot""x/2)]_(pi//2)^(pi)" "[therefore inte^(x){f(x)+f'(x)}=e^(x).f(x)]`
`=[-e^(pi)cot""pi/2]-[-e^(pi//2)cot(pi/4)]`
`=-e^(pi)xx0+e^(pi//2)xx1=e^(pi//2)`