1 Dimensionality Reducing Expansion of Multivariate Integration. - 1. 1 Darboux formulas and their special forms. - 1. 2 Generalized integration by parts rule. - 1. 3 DREs with algebraic precision. - 1. 4 Minimum estimation of remainders in DREs with algebraic precision. - 2 Boundary Type Quadrature Formulas with Algebraic Precision. - 2. 1 Construction of BTQFs using DREs. - 2. 2 BTQFs with homogeneous precision. - 2. 3 Numerical integration associated with wavelet functions. - 2. 4 Some applications of DREs and BTQFs. - 2. 5 BTQFs over axially symmetric regions. - 3 The Integration and DREs of Rapidly Oscillating Functions. - 3. 1 DREs for approximating a double integral. - 3. 2 Basic lemma. - 3. 3 DREs with large parameters. - 3. 4 Basic expansion theorem for integrals with large parameters. - 3. 5 Asymptotic expansion formulas for oscillatory integrals with singular factors. - 4 Numerical Quadrature Formulas Associated with the Integration of Rapidly Oscillating Functions. - 4. 1 Numerical quadrature formulas of double integrals. - 4. 2 Numerical integration of oscillatory integrals. - 4. 3 Numerical quadrature of strongly oscillatory integrals with compound precision. - 4. 4 Fast numerical computations of oscillatory integrals. - 4. 5 DRE construction and numerical integration using measure theory. - 4. 6 Error analysis of numerical integration. - 5 DREs Over Complex Domains. - 5. 1 DREs of the double integrals of analytic functions. - 5. 2 Construction of quadrature formulas using DREs. - 5. 3 Integral regions suitable for DREs. - 5. 4 Additional topics. - 6 Exact DREs Associated With Differential Equations. - 6. 1 DREs and ordinary differential equations. - 6. 2 DREs and partial differential equations. - 6. 3 Applications of DREs in the construction of BTQFs. - 6. 4 Applications of DREs in the boundary element method. Language: English