High-radix Booth is of great importance for speeding up multiplying by reducing the number of partial products,but it also adds great burdens on the adder of the multiplier in the aspects of timing and area.

The former reduces the number of partial product and enhances the generating speed of partial product;the latter has a regular layout because of the simplicity of the necessary interconnections between the compressors,and it reduces the spurious switching in the circuit to save power dissipation.

This paper presents a novel fast algorithm, whose partial products come from the components of the much lower rank multiplier and square.

By using differentiator series method,the formula of partial integration is obtained.

A generalization of formula of integration by parts and it s application;

This paper,by using the formula of sum-subtraction transform,improves the formula of integration by parts to get the integration by parts to further establish under the condition that derivable function u(x),v(x) in the interval and the integral ?bau(x)dv(x) still exist.

The paper mainly discusses a simply way of the subsection integral and illustrates its application with some examples.

This article intensifies and argues the median of the first median theory on integral calculus, and the second median theory on integral calculus by Abel transform and subsection integral as well.

Through the analysis of several examples for integral by parts, this paper shows that two-column differention-integral method is simple and direct, and can be widely used in the teaching of integral by parts.