  We describe a method of integration to obtain identities of the arctangent
function and show how this method can be applied to the high-accuracy
computation of the constant pi using the equation $\pi = 4 \arctan \left( 1
\right)$. Our approach combines the midpoint method with the Taylor expansion
series to enhance accuracy in the subintervals. The accuracy of this method of
integration is determined by number of subintervals $L$ and by order of the
Taylor expansion $M$. This approach provides significant flexibility in
computation since the required convergence in resulting equations can be
optimized through appropriate choices for the integers $L$ and $M$. Sample
computations are presented to illustrate that even with relatively small values
of the integers $L$ and $M$ the constant $\pi$ can be computed with high
accuracy.
