Page 39 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes r
 x  In(ax )dx
 
(iii) dv
u
dx

 sinbx
ax  
 e   or  dx
 
 cosbx 
(iv)  
dv
u
dx
Here you can consider u as ‘f ‘and v as ‘g’, as we have used f and g variables in the unit.
A general fault of those accessing integration by parts is abandoned to put a dx with the term in
both df and dg. This is a negligible point, but, for the want of comprehensiveness, it requires to
be incorporated on transitional steps.

!

Caution Keep in mind whenever we utilize integration by parts, we make use of everything
inside of the integral for f and dg that comprises the dx.

Example:
log xdx


 x  log xdx    dx log x xdx  dx
I
x 2 1 x 2
 log x   . dx
2 x 2
1 1
 x 2 log x   xdx
2 2
1 2 1 2
 x log x  x Ans .
2 4
Example:
 x cosxdx  x  cosx   ( .sin )dx
n
a
 x sinx  cosxAns .

Example:
1
x
 log xdx  log x  1dx   ( )dx
x

 x log x   dx  x log x x

 x (log 1)
Example:

 e x cosxdx


Let I = e x cosxdx

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