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