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SEC 8.2: TRIGONOMETRIC INTEGRALS
Example
Find dxx cos3 xdxxdxx coscoscos 23
xdxx cos)sin1( 2
Example
Find dxxx cossin 25
xdxxx sincos)cos1( 222
dxxxxdxxx sin cossin cossin 2425
TRIGONOMETRIC INTEGRALS
dxxx nm cossin
oddcos
oddisn
to express the remaining factors in terms of sin
dx with cos one save
x-x 22 sin1cosuse
1
2
xdxxdxx coscoscos 23
oddsin
oddism
to express the remaining factors in terms of cos
in one save s
x-x 22 cos1sinuse
1
2
dxxxx
dxxx
sin cossin
cossin
24
25
TRIGONOMETRIC INTEGRALS
odd even
sin cos
even odd
odd odd
even even
evensin
sometimes helpful to use
anglehalfuse
)2cos1(sin2
12 x-x
1
2 dxx 2
4
1 2cos1
evencos
)2cos1(cos2
12 xx
xxx 2sincossin2
1
dx sin 4
Example
Eliminating Square Roots
TRIGONOMETRIC INTEGRALS
)2cos1(cos2
12 xx
4
04cos1
dxxFind
we use the identity
to eliminate a square root.
We can use a similar strategy to evaluate integrals of the form
TRIGONOMETRIC INTEGRALS
dxxx nm sectan
Example
Find dxxx sectan 46
Example
Find dxxx sectan 35
xdxduxu 2sectan
xdxxduxu tansecsec
xx 22 tan1sec
xx 22 sec1tan
TRIGONOMETRIC INTEGRALS
dxxx nm sectan
evensec
ven eisn
to express the remaining factors in terms of tan
2ec one save s
xx 22 tan1secuse
1
2
oddtan
oddism
to express the remaining factors in terms of sec
xx tansec one save
xx 22 sec1tanuse
1
2
EXAM-2Term-082
TRIGONOMETRIC INTEGRALS
odd even
tan sec
even odd
odd odd
even even
the guidelines are not as clear-cut. We may need to use identities, integration by parts, and occasionally a little ingenuity.
tan seceven odd
TRIGONOMETRIC INTEGRALS
the guidelines are not as clear-cut. We may need to use identities, integration by parts, and occasionally a little ingenuity.
tan seceven odd
Example
Find xdx3secIf an even power of tangent appears with an odd power of secant, it is helpful to express the integrand completely in terms of sec x
Powers of sec x may require integration by parts, as shown in the following example.
TRIGONOMETRIC INTEGRALS
Example
Find xdx3sec
TRIGONOMETRIC INTEGRALS
Integrals of the form
can be found by similar methods because of the identity
dxxx nm csccot
xx 22 csccot1
REMARK
dxxx nm csccot
evencsc
ven eisn
to express the remaining factors in terms of cot
2sc one save c1
2
oddcot
oddism
to express the remaining factors in terms of csc
xx cotcsc one save
1csccotuse 22 xx
1
2xx 22 cot1csc
EXAM-2Term-122
Product of Sines and Cosines
dxnxmx cossin
dxnxmx sinsin
dxnxmx coscos
EXAM-2 Term-122
TRIGONOMETRIC INTEGRALS
Powers of tan x and sec x
Products of Sines and Cosines
Eliminating Square Roots
Powers of Sines and Cosines
TRIGONOMETRIC INTEGRALS
EXAM-2Term-092
EXAM-2Term-092
TRIGONOMETRIC INTEGRALS
TRIGONOMETRIC INTEGRALS
du
xdxxf sin)(cos
function of Sines and Cosines
du
xdxxf cos)(sin
du
xdxxf 2sec)(tan
function of tan and sec
du
xdxxxf tansec)(sec
du
xdxxf 2csc)(cot
function of cot and csc
du
xdxxxf cotcsc)(csc
EXAM-2Term-092
EXAM-2Term-092
EXAM-2Term-092
EXAM-2Term-092