Although the exact antiderivative of the simple mathematical expression ����(1+��) where ���� denotes the natural logarithm is not defined in mathematical analysis, its infinite series expansion is given by: ��-��²/2+��³/3-��⁴/4+��⁵/5-��⁶/6+… whose antiderivative is given by: ½��²-��³/2·3+��⁴/3·4-��⁵/4·5+…(-1)ⁿ��ⁿ/��(��-1)+…starting with ��=2 to infinity. Conforming to the mathematical definition of vectors, each term represents the inner scalar dot product of the vector component of the infinite vector series given by (-1)ⁿ��ⁿ and 1/ ��(1+��) whose ��th partial sum is given by ��/(1+��) with the alternating power vector series component given by ��²-��³+��⁴-��⁵+…(-1)ⁿ��ⁿ+… . Moreover, for a defined convergence (existence of a numerical sum), the theory of power series asserts that the independent variable �� must always have numerical values that lies between -1 and +1, that is bounded by the compound inequality -1<��<+1.

The history of mathematics credited the discovery of antiderivative to the German philosopher-mathematician by the name of Gottfried Wilhelm Leibniz (1646-1716). However, in its mathematical usage and terminology, the antiderivative is also commonly called the integral. Mathematicians often use the idea of integral to find the solutions of both ordinary and partial differential equations. Nonetheless, a differential equation of infinite series solutions can only differ by certain numerical constants. Fortunately, this unique concept of constants in a calculus of variation can now be used to define a principle of directional invariance for a successful quantum theory of the spacetime continuum.