For 0<α<π he value of the definite integral  ∫01 dx/x2+2xcos⁡α+1 is equal to

∫01 dxx2+2xcos⁡α+1=∫01 dx(x+cos⁡α)2+1−cos2⁡α=∫01 dx(x+cos⁡α)2+sin2⁡α=1sin⁡αtan−1⁡x+cos⁡αsin⁡α01=1sin⁡αtan−1⁡1+cos⁡αsin⁡α−tan−1⁡cos⁡αsin⁡α=1sin⁡αtan−1⁡cot⁡α2−tan−1⁡cot⁡α=1sin⁡απ2−α2−π2+α=α2sinα