Page 45 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes
Example:

dx
Integrate I  
3 4cos x

With z = tan(x/2) we obtain
2dx
I  




(1 z 2 )(3 4(1 z 2 )/(1 z 2 ))
dz
 2 
7 z 2

In (z  7)/(z  7)
  C
7
where we have used
1 1  1 1 
   
2


a  z 2 2a a z a z

 /4 dx
Task Evaluate the following integral:  .

0 1 sin x
Self Assessment
Fill in the blanks:
13. The magnificent substitution z = tan(x/2) permits alteration of any trigonometric integrand
into a ............................ one.
14. If z = tan(x/2), then dx = ............................ .
15. After the substitution z = tan(x/2) we get an ............................ that is a rational function of z.

3.3 Summary

 Integration by parts is a method depending on the product rule for differentiation, for
articulating one integral in provisions of another.
1
g
x
x
g
x
x
 By Integration by parts, we mean ( ) ( )f x  x   f ( ). ( )dx   f 1 ( ). ( )d .
 To write the result in a symmetrical form, replace f(x) by f (x) and write f (x) for g(x). Then
1 2
x
for g(x) we shall have to write f 2  ( ) dx the above equation then becomes
x
f 1  ( ) ( )dx  f 1  2 ( )    1 1 ( ) f 2  ( )dx  dx i.e. the integral of the product of two
x
x
( ) f
f
x
f
x
x
2
functions.
 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.
 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.
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