Definite Integrals, by D. Bierens de Haan*, and have found it of use for verification as well as for guidance; and many of the examples given in the following pages have been selected from it, and from its companion volume of Tables of Integrals. The latter indeed exhausts that part of the subject in its present state.

82.] The method for the evaluation of a definite integral which first presents itself, is that of deriving it from the indefinite integral when the latter can be found. The method depends on the equation (11), Art. 5; viz.,

"r′(x) dx = [r (x)] ́*;

= F(X) — F (Xo) ;

(1)

(2)

that is, the definite integral is the excess of the value of the indefinite integral when the superior limit is substituted for the variable over that when the inferior limit is substituted. The following are examples of this method.

This is a series of terms in harmonical progression: consequently

* Exposé de la Théorie des Propriétés, des Formules de Transformation, et des Méthodes d'Évaluation des Intégrales Définies; par D. Bierens de Haan. Amsterdam, C. G. Van der Post, 1860, 1862: publiée par l'Académie Royale des Sciences à Amsterdam.

the sum of such a series may be expressed as a definite integral, although it cannot be expressed in an algebraical form.

These are two remarkable instances of the mode in which definite integration may be applied to the determination of quantities apparently indeterminate. More will be said on the subject when the integrals are determined by a different process hereafter. See Art. 86.

П

2

sin ma sin nx dx = O, if m is not equal to n; (10)

(11)

(12)

The remark made at the end of Art. 6 is of great importance in reference to examples such as this and Ex. 5, viz. that the value of the infinitesimal element corresponding to the superior limit is excluded, while that corresponding to the inferior limit is included in the definite integral; for were this not the case, as

xn

(1 − x2) 1⁄2 becomes equal to ∞, when a = 1, the integrals would not satisfy the conditions, which the theory of such summation requires : but as the limit unity, being the superior limit in the above examples and that which renders infinite the infinitesimal element, is not included, the definite integrals are correct.

and therefore if n is a very large number, we have the following approximate value of π,

π = 2

2.2.4.4.6.6 ........
1.3.3.5.5.7.

;

(15)

an equivalent for which was first discovered by Dr. Wallis, and will be of considerable use in the sequel. These results are also

and all the other pairs of conjugate factors of 2"+1 will produce similar fractions, the several denominators being