8

8.110

1. Every integral of the form $\int R\left(x,\sqrt{P(x)}\right)\text{d}x$, where P(x) is a third- or fourth-degree polynomial, can be reduced to a linear combination of integrals leading to elementary functions and the following three integrals:

$\begin{array}{ccc}{\displaystyle \int \frac{\text{d}x}{\sqrt{(1-{x}^{2})(1-{k}^{2}{x}^{2})}},}& {\displaystyle \int \frac{\sqrt{(1-{k}^{2}{x}^{2})}}{(1-{x}^{2})}}\text{d}x,& {\displaystyle \int \frac{\text{d}x}{(1-n{x}^{2})\sqrt{(1-{x}^{2})(1-{k}^{2}{x}^{2})}},}\end{array}$

which are called respectively elliptic integrals of the first, second, and third kind in the Legendre normal form. The results of this reduction for the more frequently encountered ...

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