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What are integrals?

What are integrals? Here are some definitions.

Noun
  1. plural of integral
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In Abel's third paper, Solutions of some problems by means of definite integrals he gave the first solution of an integral equation.
His work on algebra was an extension of Abel's work on algebraic functions and their integrals.
His interpolation used Kepler's concept of continuity, and with it he discovered methods to evaluate integrals.
Galois, after reading Abel and Jacobi's work, worked on the theory of elliptic functions and abelian integrals.
Both the peak values and the integrals under the characteristic fluorescence curves were measured.
In 1823 Abel published papers on functional equations and integrals in a new scientific journal started up by Hansteen.

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