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A compilation of a list of integrals (Integraltafeln) and techniques of integral calculus was published by the German mathematician Meyer Hirsch in 1810. These tables were republished in the United Kingdom in 1823. More extensive tables were compiled in 1858 by the Dutch mathematician David de Bierens de Haan. A new edition was published in 1862. These tables, which contain mainly integrals of elementary functions, remained in use until the middle of the 20th century. They were then replaced by the much more extensive tables of Gradshteyn and Ryzhik. In Gradshteyn and Ryzhik, integrals originating from the book by de Bierens are denoted by BI. Not all closed-form expressions have closed-form antiderivatives; this study forms the subject of differential Galois theory, which was initially developed by Joseph Liouville in the 1830s and 1840s, leading to Liouville's theorem which classifies which expressions have closed form antiderivatives. A simple example of a function without a closed form antiderivative is e−x2, whose antiderivative is (up to constants) the error function. Since 1968 there is the Risch algorithm for determining indefinite integrals that can be expressed in term of elementary functions, typically using a computer algebra system. Integrals that cannot be expressed using elementary functions can be manipulated symbolically using general functions such as the Meijer G-function. |
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