4.3. INTEGRATING TRIG FUNCTIONS 141
4.3 INTEGRATING TRIG FUNCTIONS
For completeness we start with a Knowledge Box of the integrals of trigonometric functions
we have already obtained. Most of this section is about how to deal with products of powers of
trig functions, and these integrals are building blocks.
Knowledge Box 4.6
Trig-related integral forms
•
Z
sin.x/ dx D cos.x/ C C
•
Z
cos.x/ dx D sin.x/ CC
•
Z
tan.x/ dx D ln jsec.x/j C C
•
Z
sec.x/ dx D ln jtan.x/ C sec.x/j CC
•
Z
sec
2
.x/ dx D tan.x/ CC
•
Z
csc
2
.x/ dx D cot.x/ C C
•
Z
sec.x/ tan.x/ dx D sec.x/ CC
•
Z
csc.x/ cot.x/ dx D csc.x/ CC
Odd powers of sine and cosine
Functions of this sort are integrated by exploiting the Pythagorean identity. We leave
one power of the trig function to be d U and transform the remaining even powers via one of
142 4. METHODS OF INTEGRATION I
cos
2
.x/ D 1 sin
2
.x/
or
sin
2
D 1 cos
2
.x/
as appropriate, transforming the problem into an integral of a polynomial function.
Example 4.47 Find
Z
cos
5
.x/ dx:
Solution:
Z
cos
5
.x/ dx D
Z
cos
2
.x/
2
cos.x/ dx
D
Z
1 sin
2
.x/
2
cos.x/ dx
Let u D sin.x/, du D cos.x/ dx
D
Z
.1 u
2
/
2
du
D
Z
1 2u
2
C u
4
du
D u
2
3
u
3
C
1
5
u
5
C C
D sin.x/
2
3
sin
3
.x/ C
1
5
sin
5
.x/ C C
˙
Odd powers of sine work the same way, but with the other function in a starring role.
4.3. INTEGRATING TRIG FUNCTIONS 143
Even powers of sine and cosine
ese are much messier and rely on the power reduction identities:
sin
2
.x/
D
1 cos.2x/
2
cos
2
.x/ D
1 Ccos.2x/
2
ey can always be used to transform an even power into a bunch of much lower powers, both
odd and even. is means that even powers of trig functions often end up as furballs, but they
can be done with patience and persistence.
Example 4.48 Compute
Z
sin
4
.x/ dx
Solution:
Z
sin
4
.x/ dx D
Z
sin.x/
2
2
dx
D
Z
1 cos.2x/
2
2
dx
D
1
4
Z
1 2 cos.2x/ Ccos
2
.2x/
dx
D
1
4
Z
1 2 cos.2x/ C
1 Ccos.4x/
2
dx
D
1
8
Z
.
2 4 cos.2x/ C1 C cos.4x/
/
dx
D
1
8
Z
.
3 4 cos.2x/ Ccos.4x/
/
dx
D
1
8
3x 4
1
2
sin.2x/ C
1
4
sin.4x/
C C
D
3
8
x
1
4
sin.2x/ C
1
32
sin.4x/ CC
˙
144 4. METHODS OF INTEGRATION I
Even powers of secant
ese are pretty easy, again by hitting them with Pythagorean identities. Save one sec
2
.x/ to
be du and apply
sec
2
.x/ D 1 C tan
2
.x/
to set up the u-substitution u D tan.x/.
Example 4.49 Compute
Z
sec
8
.x/ dx
Solution:
Z
sec
8
.x/ dx D
Z
sec
2
.x/
3
sec
2
.x/ dx
D
Z
1 Ctan
2
.x/
3
sec
2
.x/ dx
Let u D tan.x/, du D sec
2
.x/ dx
D
Z
1 Cu
2
3
du
D
Z
1 C3u
2
C 3u
4
C u
6
du
D u Cu
3
C
3
5
u
5
C
1
7
u
7
C C
D tan.x/ C tan
3
.x/ C
3
5
tan
5
.x/ C
1
7
tan
7
.x/ C C
˙
4.3. INTEGRATING TRIG FUNCTIONS 145
Odd powers of secant
ese are typically a nightmare!
Example 4.50 Compute
Z
sec
3
.x/ dx
Solution:
Z
sec
3
.x/ dx D
Z
1 Ctan
2
.x/
sec.x/ dx
D
Z
sec.x/ dx C
Z
tan
2
.x/ sec.x/ dx
D ln jsec.x/ Ctan.x/j C
Z
tan.x/ sec.x/ tan.x/ dx
U D tan.x/I dV D sec.x/ tan.x/ dx Use parts
du D sec
2
.x/ dxI V D sec.x/
D ln jsec.x/ Ctan.x/j Csec.x/ tan.x/
Z
sec
3
.x/ dx
2
Z
sec
3
.x/ dx D ln jsec.x/ C tan.x/j Csec.x/ tan.x/ Circularize!
Z
sec
3
.x/ dx D
1
2
.
ln jsec.x/ Ctan.x/j Csec.x/ tan.x/
/
C C
As with all circular integrations by parts, we need to remember to finish with a CC . Higher
odd powers are worse.
˙
Powers of tangent
Even powers of tangent can be transformed into even powers of secant by using
tan
2
.x/ D sec
2
.x/ 1.
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