Page 12 - DMTH202_BASIC_MATHEMATICS_II

Unit 1: Integration

Before starting, however, let us remind some helpful trigonometric identities. Notes
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2
2
The first is the Pythagorean Identity: sin  + cos  = 1. If we divide all terms by cos , then we
have:
2
2
sin  cos  1
2
2
   tan   1 sec  .

2
2
cos  cos  cos 2
The well-known trigonometric identities
a 2  a 2 sin 2   t a 2 cos 2   t   1
a 2  a 2 tan 2   t a 2 sec 2   t   2
a 2 sec 2   t  a 2 tan 2   t   3
may be used to eradicate radicals from integrals. Particularly when these integrals entail a  x 2
2
2
2
and x  a .
1. For a  x 2 set x  a sin   t . In this case we converse regarding sine-substitution.
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2. For a  x 2 set x  a tan   t . In this case we converse regarding tangent-substitution.
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3. For x  a set x  a sec   t . In this case we converse regarding secant-substitution.
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2
2
2
2
2
The expressions a  x and x  a should be observed as a constant plus-minus a square of a
2
function. Here, x displays a function and a a constant. For example, x  2x  3 can be observed as
one of the two previous expressions. Certainly, if we complete the square we obtain
2
2
3
x  2x   x   1  2
1
2
where a = 2. Thus from the above substitutions, we will set x   2 tan   t .
The following examples exemplify how to apply trigonometric substitutions:
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Example: Find x 3 4 x dx


Solution:
It is simple to observe that sine-substitution is the one to use. Set x  2sin   t or equivalently
t  sin  1  /2x . Then dx  2cos   t dt which provides us

2
 x 3 4 x dx   8sin 2   4 4sint  2  2 cost   .t dt

Easy calculations provide
2


 x 3 4 x dx  32 sin 2  cost 2   .t dt
Technique of integration of powers of trigonometric functions give
 sin 3  cost 2   t dt   1 cos 2   cost 2  sint   t dt

which recommends the substitution v  cos   t . Hence dv   sin   t dt which implies

v 3 v 5
2
2
2
2
 1 cos   cost   sint   t dt     1 v v dv      . C
3 5
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