6.4 Integration by parts 173
6.4 Integration by parts
In the integral
the integrand is the product of two quite different types of function; the polynomial x
and the trigonometric functioncos 1 x. The value of the integral is
y 1 = 1 x 1 sin 1 x 1 + 1 cos 1 x 1 + 1 C
as can be verified by differentiation:
= 1 (sin 1 x 1 + 1 x 1 cos 1 x) 1 + 1 (−sin 1 x) 1 = 1 x 1 cos 1 x
The product rule has been used to differentiatethe productx 1 sin 1 x, and the method of
integration by parts is used to integrateproducts of this type. In general, lety 1 = 1 uv,
where uand vare functions of x. Then
(6.12)
and, integrating both sides of this equation with respect to x,
(6.13)
The left-hand side is equal toy 1 = 1 uvby definition, and the equation can be rearranged as
(6.14)
This equation, the inverse of the product rule, is the rule of integration by parts. Given
an integrand likex 1 cos 1 x, one of the factors is identified with uin (6.14), the other with
dv 2 dx. For example, let
u 1 = 1 x,
Then
v 1 = 1 sin 1 x
du
dx
=, 1
d
dx
x
v
=cos
ZZu
d
dx
dx u
du
dx
dx
v
=−vv
ZZ Z
dy
dx
dx u
d
dx
dx
du
dx
=+dx
v
v
dy
dx
u
d
dx
du
dx
=+
v
v
dy
dx
d
dx
xx
d
dx
= x
()+
()sin cos