The first volume, on indefinite integrals, was published by Notdruck ( Braunschweig ) in 1944 and by Springer in 1949 . In 1950, the second volume containing definite integrals appeared.
22.
Since the concept of an antiderivative is only defined for functions of a single real variable, the usual definition of the indefinite integral does not immediately extend to the multiple integral.
23.
When I input Sqrt [ 2 * Pi * x ] * x ^ x * Exp [-x ] I get basically the same indefinite integral back, not an error message.
24.
There is a differential Galois theory, but it was developed by others, such as Picard and Vessiot, and it provides a theory of quadratures, the indefinite integrals required to express solutions.
25.
There are yet other cases ( such as the Gaussian integral ) where definite integrals can be evaluated exactly without numerical methods, but indefinite integrals cannot, for lack of an elementary antiderivative.
26.
Finding an indefinite integral of a function f ( x ) is the same as solving the differential equation \ frac { dy } { dx } = f ( x ).
27.
Risch called it a decision procedure, because it is a method for deciding whether a function has an elementary function as an indefinite integral, and if it does, for determining that indefinite integral.
28.
An attempt to introduce an indefinite integral as the inverse of differentiation is bound to fail as the second derivative of any Grassmann function vanishes, so that the inverse operation does not exist.
29.
Risch called it a decision procedure, because it is a method for deciding whether a function has an elementary function as an indefinite integral, and if it does, for determining that indefinite integral.
30.
You shouldn't really say " indefinite integral " in this context since, strictly speaking, the integral should have lower and upper limits ( respectively, the initial and final positions of the object being moved ).
