Calculus: Analytic Geometry and Calculus, with Vectors

9.5 Integration by parts 519

are functionshaving continuous derivatives over intervals appearing in
our work, theformula which can be put in the forms

dxuv= udx+vdx, d(uv) = udv+vdu

udv duv du

d x

_

dx -

v

dx'

udv = d(uv) -vdu

implies the formula which can be put in they forms

(9.51) J udvdx = uv - J vdx dx, f u dv = uv - J v du

(9.52) J u(x)v(x) dx = u(x)v(x) -

J

v(x)u'(x) dx

which involve the notations of Leibniz and Newton. We can prefer

to use (9.52) when meanings of symbols are being explainedand to use

(9.51) when problems are being solved and the abbreviated notation

expedites our work without confusing us.

We have already seen some of the reasons why the formula (9.51) or

(9.52) for integration by parts is useful. As was pointed out following

(8.483), efficient use of the formula is made by writing

(9.53) u = u(x), dv = v'(x) dx

(9.54) du = u(x) dx, v = fv'(x) dx = v(x),

where u(x) and v'(x) are chosen in such a way that the product u(x)v'(x)

is the integrand in the integral we wish to study. The integral of the

product of the things in (9.53) is then the product of the things on the

main diagonal minus the integral of the product of the things in (9.54).

The formula for integration by parts has so many applications that it

is quite hopeless to undertake to tell when and how it is useful. In

many (but not all) situations, the formula is useful when u and v' are

chosen in such a way that fv'(x) dx is an elementhry integral and the

integral fv(x)u'(x) dx is simpler than the integral fu(x)v(x) dx. Our

examples and problems will provide some ideas and information. Mean-

while, our guiding principle merits repetition. If we want to learn some-

thing about an integral and other methods fail to be helpful, we try

integration by parts. Before turning to examples, we make a final

observation. In order to apply the formula (9.52) for integration by

parts, we need just one pair of functions u and v for whichu(x)v'(x) is a

given integrand. It is therefore not necessary to insert an added con-

stant ci of integration when we write a function v whosederivative is v'.

One who wishes to do so may see what happens when we replace v = -e-

by v = -e-x + ci in the following example. There are relatively few