Page 44 - DMTH202_BASIC_MATHEMATICS_II

Unit 3: Integration by Parts

10. Integration by parts permits us to integrate numerous products of functions of x. Notes
11. A general mistake of those accessing integration by parts is neglected to put a dx with the
term in both df and dg.

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

3.2 The Substitution z = tan(x/2)

The magnificent substitution z = tan(x/2) permits alteration of any trigonometric integrand into
a rational one.
Let us consider our integrand as a rational function of sin(x) and cos(x).

After the substitution z = tan(x/2) we get an integrand that is a rational function of z, which can
then be assessed by partial fractions.
Theorem:

If z = tan(x / 2), then
2dz
dx  ,
1 z 2


1 z 2
cos x  ,
1 z 2

and
2z
sin x
1 z 2

and any rational function of xdx turns out to be a rational function of zdz.
Proof.
By the rules for differentiation we contain, for z = tan(x/2),

2
sec ( /2) 1 z 2

x
dz  dx  dx
2 2
The angle addition formulae provide us:
x
x
sin x  2 sin( /2)cos( /2)
2
x
x
 2 tan( /2)cos ( /2)
2z

2
sec ( /2)
x
2z


1 z 2
2
2
x
x

cos x  cos ( /2) sin ( /2)
2
x

 (1 z 2 )cos ( /2)
(1 z 2 )


2
sec ( /2)
x
(1 z 2 )



(1 z 2 )
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