Evaluation of a Subclass Kp,r=∫xp(1+x2)rdx of the Binomial Integral

Employing the fundamental techniques of substitution, or change of variables, and integration by parts, the discussion in this paper focuses on obtaining explicit formulas for the integral Kp,r=∫xp1+x2rdx for p∈Z and r∈12Z. As the case where r is a non-negative integer is rather simple and obvious, we concentrate mainly on the cases where r is a negative integer or an odd multiple of 12 except when consideration of other cases is appropriate.