Page 38 - DMTH202_BASIC_MATHEMATICS_II

Unit 3: Integration by Parts

To write the result in a more symmetrical form, replace f(x) by f (x) and write f (x) for g(x). Then Notes
1 2
x
for g(x) we shall have to write f 2  ( )dx the above equation then becomes.
f
x
f 1  ( ) ( )dx  f 1  2 ( )   1 f 1 ( ) f 2  ( )dx  dx
x
( ) f
x
x
x
x
2
i.e. the integral of the product of two functions = first function  integration of second – Integral
of diff. {Coeft. Of first integral of second}
Integration with the help of this rule is called integration by parts. The success of the method
depends upon choosing the first function in such a way that the second term on the right hand
side may be easy to the product is regarded as the first function.
!
Caution Case must be taken in choosing the first function.
Integration by parts permits us to integrate numerous products of functions of x. We consider
one aspect in this product to be f (this also occurs on the right-hand-side, together with df/dx).
The other aspect is considered to be dg/dx (on the right-hand-side only g occurs – i.e. the other
factor integrated concerning x).

Notes It is important to note that

1. Unity may be taken in certain cases as one of the functions.
2. The formula of integration by parts can be applied more than once if necessary.

Did u know? If the integral on the right-hand side reverts to the original form, the value of
the integral can be immediately inferred by transposing the forms to the left-hand side.

3.1.1 Usage

Integration by parts is used when we observe two dissimilar functions that don’t appear to be
associated to each other via a substitution.

2
x

2
Example: 2xe dx does not need integration by parts as 2x is the derivative of x .
Usually, one will observe a function that contains two dissimilar functions. We emphasize here
four dissimilar types of products for which integration by parts can be accessed (in addition to
which factor to tag f and which one to tag dg/dx). These are:
 sinbx
n  
 x   or  dx
 cosbx 
 
(i)  
dv
u
dx
n
ax
 x  e dx
 
(ii) dv
u
dx

LOVELY PROFESSIONAL UNIVERSITY 33