Page 52 - Standard Handbook Of Petroleum & Natural Gas Engineering

Differential and Integral Calculus 41

or
u dv = d(uv) - v du
and by integrating

ju dv = uv - jv du
where Iv du may be recognizable as a standard form or may be more easily
handled than ju dv.
Integration by Transformation may be useful when, in certain cases, particular
transformations of a given integral to one of a recognizable form suggest
themselves.
For example, a given integral involving such quantities as
Ju2_a2, JiFT-2, or J2TF

may suggest appropriate trigonometric transformations such as, respectively,
u = a csc 8,
u = a tan 8,
or
u = a sin 8

Integration by Partial Fractions is of assistance in the integration of rational
fractions. If
ax+b - ax+b =-+- B
A
-
x2+px+q (x-a)(x-P) x--01 x-P
where A + B = a
AP + Ba = -b
and A and B are found by use of determinants (see “Algebra”), then

(ax + b)dx
Alog(x-a)+Blog(x-P)+C

Integration by Tables is possible if an integral may be put into a form that can
be found in a table of integrals, such as the one given in Table 1-7. More
complete tables may be found in Bois, “Tables of Indefinite Integrals,” Dover,
and in others.

Definite integrals
The Fundamental Theorem of Calculus states that if f(x) is the derivative of F(x)
and if f(x) is continuous in the interval [a,b], then

ja?(x)dx = F(b)-F(a)
(fext continued on page 44)