Page 42 - DMTH202_BASIC_MATHEMATICS_II

Unit 3: Integration by Parts

Deduce Notes
x
df  (n  2)tan sec n 2 x dx ;g  tan (except for n  2, when df  0).
x
Use
2
2
tan x  sec x  1
to rewrite
n
g df  (n  2) sec x  sec n  2  x n 2
Get

 
G ( ) sec n 2 x tan x   n  2  G n   G n   2

n

Reorganize
   n 
n  1 G sec n 2 x tan x   n  2   G n   2

Iterate this to get
 1 n  2 (n  2)(n  4)  
2
4
n
G ( )  sec n 2 x tan  x   cos x cos x   
 n  1 (n  1)(n  3) (n  1)(n  3)(n  5)   
When n is even this discontinues automatically; when n is odd, the outcome is in our table of
simple trigonometric integrals for n = 1:
G (1) In (sec x  tan  ) x

Example:
sqrt(1+x )
2
First method
2
To evaluate:  1 x dx

Substituting
x = tan y
You obtain

3
2

 1 x dx   sec y dy
 G (3)
Using the above example you obtain

y
sec y tan y  In (sec y  tan )
G (3) 
2

2

x 1 x  In x  1 x 2 

G (3) 
2
Alternate method by substitution
2
To evaluate:  1 x dx

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