Page 40 - DMTH202_BASIC_MATHEMATICS_II

Unit 3: Integration by Parts

 e n sinx   e n sinxdx Notes
n

 e n sinx   ( cos )e  x   e n ( cos )dx 
x
 e n sinx e n cosx   e n cosxdx

 e n (sinn  cos ) I

n
1
x
 e n (sinn  cos )Ans .
2
Example:
x


1
1

 1. sin x dx  sin x x   dx
1 x 2

x
1

 x sin x   dx
1 x 2

1 2x

1
x sin x   dx
2 1 x 2

1

 x sin x  In  1- x 2   C
Example:
 x 2 cos2xdx
2 sin2x sin2x
x
 x   2 . dx
2 2

x 2  ( cos2 ) ( cos2 )
x

x
 sin2x   x    dx
2  2 2 
1 1 sin2x
 x 2 sin2x  x cos2x 
2 2 4
Example:
 In x dx
Set
f = ln x; dg = dx
Deduce

dx
df  ;g  x C

x
Any value of C can be used here.
Here and in the other examples, we select C = 0.
Get

 In x dx  x In x   dx
 x (In x  1)

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