Um caso especial de integrais binômias

Nogueira, Iszaias Rangel

doi:10.1590/S0071-12761960000100002

Resumo

The present paper shows that the sum of two binomial integrals, such as A ∫ x p (a + bx q)r dx + B ∫ x p (a + bx q)r dx, where A and B are real constants and p, q, r are rational numbers, can, in special cases, lead to elementary integrals, even if each by itself is not elementary. An example of the case considered is given by the integral ∫ x _____-___ 3 dx = 1/2 ∫ x-½ (x - 1)-⅓ dx - 6 √ x ³√(x - 1)4 = 1/3 ∫ x-½ (x - 1)-¾ dx On the rigth hand side of the last equality both integral are not elementary. But the use of integration by parts of one of them leads to the solution: ∫ x _____-___ 3 dx = x½ (x - 1)-⅓ + C. 6 √ x ³√(x - 1)4

Escola Superior de Agricultura «Luiz de Queiroz»

SUMMARY

The present paper shows that the sum of two binomial integrals, such as

A ∫ x^p (a + bx^q)^r dx + B ∫ x^p (a + bx^q)^r dx,

where A and B are real constants and p, q, r are rational numbers, can, in special cases, lead to elementary integrals, even if each by itself is not elementary. An example of the case considered is given by the integral

∫ x _____-___ 3 dx = 1/2 ∫ x-^½ (x - 1)^-⅓ dx -

6 √ x ³√(x - 1)⁴ = 1/3 ∫ x-^½ (x - 1)^-¾ dx

On the rigth hand side of the last equality both integral are not elementary. But the use of integration by parts of one of them leads to the solution:

∫ x _____-___ 3 dx = x^½ (x - 1)^-⅓ + C.

6 √ x ³√(x - 1)⁴

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LITERATURA CITADA

Recebido para publicação em 23 de março de 1960.

CATUNDA, Omar - 1953 - Curso de Análise Matemática - III Parte. Editôra Bandeirantes, S. Paulo.
POTRON, M. L' Abbé - L' Integrale de Différentielle Binome. Gauthier - Villars, Paris.