02 trigonometric integrals (gino edit1)

8/11/2019 02 Trigonometric Integrals (Gino Edit1)

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Trig Integrals Exercises

Techniques of IntegrationTrigonometricIntegrals

Mathematics 54Elementary Analysis 2

Institute of Mathematics

University of the Philippines-Diliman

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Trigonometric Integrals

We will study techniques for evaluationg integrals of the form

sinm

xcosnx dxsecmxtannx dxcscmxcotnx dx,

where either mor nis a positive integer.

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Trigonometric Integrals

Recall:

Some identities:

sin2 x+cos2 x= 1

1+ tan2 x= sec2 x

1+cot2 x=csc2 x

sin2 x=1cos2x

2

cos2 x=1+cos2x

2

Derivatives:

Dx(sin x) =cos x

Dx(cos x) = sin x

Dx(tan x) =sec2

x

Dx(cot x) =csc2 x

Dx(sec x) =sec xtan x

Dx(csc x) =csc xcot x

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Trigonometric Integrals: Some examples from Math 53

Examples:Find the following antiderivatives.

1

sin7 x cos x dx=

u

7du=

u8

8+C=

sin8 x

8+C

Let u= sin x= du= cos x dx

2

tan3 xsec2 x dx=

u

3du=

u4

4+C=

tan4 x

4+C

Let u= tan x= du= sec2 x dx

3

cot2 x dx=

(csc2 x1) dx=cot xx+C

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Consider:

cos3 xsin2 x dx=?

cos

3

xsin2

x dx =

cos2

xsin2

xcos x dx

=

1sin2 x

sin2 x cos x dx

=

sin2 xcos x dx

sin4 xcos x dx

let u= sin x du= cos xdxcos3 xsin2 x dx =

u

2du

u

4du

=1

3

u3

1

5

u5+C=

1

3

sin3 x

1

5

sin5 x+C

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T i I l E i


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Some guidelines for evaluating trigonometric integrals

Some guidelines for evaluating integrals of the form

sinmxcosnx dx,

secmx tannx dx,

cscmxcotnx dx,


G1. Try to split off a factor equal to any of the derivatives of the six

trigonometric functions (e.g. cos3 x sin2 x= cos2 x sin2 xcos x).

Express the rest of the factors in terms of the corresponding

trigonometric function using the identities, avoiding the

introduction of any or additional radicals.

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T i I t l E i


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Some guidelines for evaluating trigonometric integrals

Some guidelines for evaluating integrals of the form

sinmxcosnx dx,

secmx tannx dx,

cscmxcotnx dx,


G1. split off: express the rest in terms of:

sin x cos x

cos x sin x

sec2 x tan x

csc2

x cot xsec xtan x sec x

csc xcot x csc x

Then expand, substitute, and integrate.

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Example.

Evaluate sin3 x dx.Note that

sin3 x = sin2 xsin x

= (1cos2 x) sin x

Thus,

sin3 x dx=

1 cos 2x

sin x dx.

Let u= cos x, du=sin x dx. Therefore,

sin3 x dx=

1u2

du=u+ 1

3u

3+C=cos x+ 1

3cos3 x+C

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Example.

Evaluate

tan x sec4 x dx.

tan xsec4 x dx =

tan xsec2 xsec2 x dx

=tan x(1+ tan2 x) sec2 x dx

=

tan x+

tan5 x

sec2 x dx

let u= tan x du= sec2 x dx=u1/2+u5/2dx=

2

3u

3/2 +2

7u

7/2 +C

=2

3

tan3 x+

2

7

tan7 x+C

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Example.

Evaluate csc6 x dx.

csc6 x dx =

(csc2 x)2 csc2 x dx

=

1+cot2 x

2 csc2 xdx

=

(1+2cot2 x+cot4 x)csc2 x dx

let u= cot x du=csc2 x dx

csc6 x dx =(1+2u2 +u4) du=

cot x+

2cot3 x

3 +

cot5 x

5

+C

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Example.

Evaluate tan3 xsec2 x dx.

tan3 xsec2 x dx =

tan2 xsec xsec xtan x dx

=

sec2 x1

sec xsec xtan x dx

=

sec3 xsec x

sec x tan x dx

let u= sec x, du= sec xtan x dxtan3 xsec2 x dx =

u

3u

du

=1

4sec4 x 1

2sec2 x+C

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g g

Example.

Evaluate tan3xdx.

G2. If the integrand involves only positive integer factors of tanx

or cotx, with exponents greater than 2, split off a factor of

tan2 xor cot2 x, apply the identities

tan2 x= sec2x1

cot2 x= csc2 x1

and use the substitution u= tanxor u= cotx.

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Example.

Evaluate cot4 3x dx.

cot2 3xcot2 3x dx =

cot2 3x

csc2 3x1

dx

= cot2 3x csc2 3xcot2 3x dx

=

cot2 3x csc2 3xcsc2 3x+1

dx

=

cot2 3x csc2 3x

dx+

1

3cot3x+x+C

let u= cot3x, du=3csc2

3x dxcot4 3x dx = 1

3

u

2du+

1

3cot 3x+x+C

= 19

u3+

1

3cot3x+x+C

= 1

9 cot3 3x+

1

3 cot3x+x+C 14/26



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Example.

Evaluate sin2

x dx.

G3. If the integrand involves only even powers of sin x and/or cos x,

use the identities

sin2 x= 1cos2x2

cos2 x=1+cos2x

2

to reduce the exponent.

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Example.

Evaluatesin2xdx.

sin2xdx =

1cos2x

2dx

=12

(1cos2x)dx

=1

2

x

1

2sin2x

+C

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Example.

Evaluate sin2

xcos4 x dx.

sin2 xcos4 x dx=

sin2 x(cos2 x)2dx

=1cos2x2

1+cos2x2

2

dx

=

1cos2x

2

1+cos2x

2

2dx

=1

8

1+cos2xcos2 2xcos3 2x

dx

=18

1+cos2x

1+cos4x

2

(1sin2 2x)cos2x

dx

=1

8

x+

sin2x

2 1

2

x+

sin4x

4

1

2

sin2x sin

3 2x

3

+C

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Example.

Evaluate sec3

x dx.

Note that sec3 x= sec xsec2 x.

G4. If the integrand involves only odd powers of sec x or csc x,

split off a factor of sec

2x

or csc

2x

and apply IBP.

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Example.

Evaluate sec3

x dx.

Note that sec3 x= sec xsec2 x. By IBP,

u= sec x , dv= sec2 x dx

du= sec xtanx dx , v= tan x

sec3 x dx = sec xtan x

tan x(sec xtan x) dx

= sec xtan x

tan2 xsec x dx

= sec xtan x

(sec2 x

1)sec x dx

sec3 x dx = sec xtan x

sec3 x dx+

sec x dx

2

sec3 xdx = sec xtan x+ ln |sec x+ tan x|+C

sec3 xdx =

1

2(sec xtan x+ ln |sec x+ tan x|)+C

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Example.

Evaluate tan2

xsec x dx.

G5. If the exponent for tan x is even and the exponent of secx is

odd, express the integrand in terms of secx only.

If the exponent for cot x is even and the exponent of csc x is

odd, express the integrand in terms of cscx only.

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Consider:

cos3xcos5xdx = ?

Recall. Product to Sum Formula

sinmxcosnx =1

2

[sin(m+n)x+sin(mn)x],

cosmxcosnx =1

2[cos(m+n)x+cos(mn)x].

sinmxsinnx = 1

2[cos(m+n)xcos(mn)x],

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Example.

Evaluate

cos3xcos5x dx.

cos3x cos5x dx =

1

2

(cos(3x+5x)+cos(3x5x)) dx

=1

2

(cos8x+cos2x) dx

=1

2

1

8

sin8x+1

2

sin 2x+C=

1

16sin8x+

1

4sin2x+C

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G id li f l i i i i l


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Guidelines for evaluating trigonometric integrals

1. Try to split off a factor equal to any of the derivatives of the six

trigonometric functions. Express the rest of the factors in terms

of the corresponding trigonometric function using the identities,

avoiding the introduction of any or additional radicals.

2. If the integrand involves only positive integer factors of tan x or

cot x, with exponents greater than 2, split off a factor of tan2

x orcot2 x, apply the identities

tan2 x= sec2 x1, cot2 x= csc2 x1,and use the substitution u= tan x or u= cot x.

3. If the integrand involves only even powers of sin xand/or cos x,use the identities

sin2 x=1cos2x

2, cos2 x=

1+cos2x

2

to reduce the exponent.

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G id li f l i i i i l


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Guidelines for evaluating trigonometric integrals

4. If the integrand involves only odd powers of sec x or csc x,

split off a factor of sec2 x or csc2 x and apply IBP.

5. If the exponent for tan x is even and the exponent of secx is

odd, express the integrand in terms of secx only.

If the exponent for cot x is even and the exponent of csc x is

odd, express the integrand in terms of cscx only.

6. Apply the product to sum identities for integrands of the form

sinmx

cosnx

, cosmx

cosnx

, or sinmx

sinnx

, m

,nR.

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Exercises

Evaluate the following integrals.

1

cos3 x

sin xdx

2csc4 x

cot2 x dx

3

cos 4xcos 3x dx

4 tan3(ln x)sec8(ln x)

x

dx

5

10

sin2xcos2x dx

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02 trigonometric integrals (gino edit1)

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