I =

π

∫

0

xisnx

1+cos2x

dx ... (i)
I =

π

∫

0

π−xsinπ−x

1+cos2xπ−x

dx
I =

π

∫

0

π−xsinx

1+cos2x

dx
I =

π

∫

0

πsinx

1+cos2x

dx -

π

∫

0

xsinx

1+cos2x

dx ... (2)
Adding (1) and (2), we get:
2I =

−1

∫

1

π−dt

1+t2

2I = - π

−1

∫

1

(

1

1+t2

) dt
2I = - π |tan−1t|1−1
2I = π [tan−11−tan−11 - 1]
2I = π (

π

4

−(−

π

4

))
2I =

π2

2

∴ I =

π2

4