TRIGONOMETRIC SUBSTITUTIONS

Section 8.4b

How do we evaluate this integral?3

29

x dx

x

Trigonometric Substitutions

These trigonometric substitutions allow us to replacebinomials of the form

2 2a x 2 2a x 2 2x aby single squared terms, and thereby transform a numberof integrals (like the one above) into ones we can evaluatedirectly or find in a table of integrals.

The most common substitutions are based on the followingreference triangles:

Trigonometric Substitutions

2 2a xx

a

tanx a

2 2 seca x a

With tan ,x a 2 2 2 2 2tana x a a 2 21 tana

2 2seca

The most common substitutions are based on the followingreference triangles:

Trigonometric Substitutions

ax

2 2a x

sinx a

2 2 cosa x a

With sin ,x a 2 2 2 2 2sina x a a 2 21 sina

2 2cosa

The most common substitutions are based on the followingreference triangles:

Trigonometric Substitutions

x 2 2x a

a

secx a 2 2 tanx a a

With sec ,x a 2 2 2 2 2secx a a a 2 2sec 1a

2 2tana

1. replaces with

Trigonometric Substitutionstanx a 2 2a x 2 2seca

2. replaces withsinx a 2 2a x 2 2cosa 3. replaces withsecx a 2 2x a 2 2tana

Also, we want any substitution to be reversible so we canchange back to the original variable afterward. For example:

tanx a requires 1tanx

a

with

2 2

Essentially, keep positiveswith any absolute values…

Trigonometric Substitutions

Evaluate3

29

x dx

x

Set 3sinx 3cosdx d

2 29 9cosx

32

3sin 3cos

9cos

d

327sin 3cos

3cos

d

327 sin d 227 1 cos sin d 227 sin sin cos d

327cos 9cos C

3 x

2 23 x

Trigonometric Substitutions

Evaluate3

29

x dx

x

Set 3sinx 3cosdx d

2 29 9cosx 327cos 9cos C

32 29 9

27 93 3

x xC

3 x

2 23 x

3 22

29

9 93

xx C

Trigonometric Substitutions

Evaluate2 9

dx

x

3secx 3sec tandx d

2 29 9 tanx

2

3sec tan

9 tan

d

3sec tan

3tan

d

sec d ln sec tan C Appendix A7, Formula 88 (p.631)

x2 23x

3

2 9ln3 3

x xC

Trigonometric Substitutions

Evaluate2 2 1

dx

x x

tanx 2secdx d

2 21 secx 2

2 2

sec

tan sec

d

2

2

sec

tan sec

d

2

sec

tan

d

2

2

cos

cos sin

d

2

cos

sin

d

2sin cos d

1sin C

Trigonometric Substitutions

Evaluate2 2 1

dx

x x

tanx 2secdx d

2 21 secx

1sin C

2 21 x x

1

csc C

21 xC

x