Trigonometric SubstitutionIntegrals involving 2 ...

Trigonometric Substitution Integrals involving

a2 - x2

Integrals

involving

p x

2

+

a2

Integrals involving

x2 - a2

Trigonometric Substitution

To solve integrals containing the following expressions;

p a2 - x2

p x2 + a2

p x2 - a2,

it is sometimes useful to make the following substitutions:

Expression q

a2 - x2

pa2 + x2

q x2 - a2

Substitution

x = a sin ,

-

2

2

or

= sin-1

x a

x = a tan ,

-

2

2

or

= tan-1

x a

x = a sec ,

0

<

2

or

<

3 2

or

= sec-1

x a

Identity 1 - sin2 = cos2 1 + tan2 = sec2

sec2 - 1 = tan2

Note The calculations here are much easier if you use the substitution in

reverse:

x

=

a sin

as

opposed

to

= sin-1

x a

.

Annette Pilkington

Trigonometric Substitution

Trigonometric Substitution Integrals involving

a2 - x2

Integrals

involving

p x

2

+

a2

Integrals involving

Integrals involving a2 - x2

x2 - a2

We make

a2 - x2

the substitution

=

p a2

-

a2

sin2

x

= =

a sin , a| cos |

-

2

=a

2

cos (since

, dx = a cos d,

-

2

2

by choice. )

Example

Z x3

dx

4 - x2

Annette Pilkington

Trigonometric Substitution

Trigonometric Substitution Integrals involving

a2 - x2

Integrals

involving

p x

2

+

a2

Integrals involving

Integrals involving a2 - x2

x2 - a2

We make

a2 - x2

the substitution

=

p a2

-

a2

sin2

x

= =

a sin , a| cos |

-

2

=a

2

cos (since

, dx = a cos d,

-

2

2

by choice. )

Example

Z x3

dx

4 - x2

Let x = 2 sin , dx = 2 cos d,

4

-

x2

=

p 4

-

4

sin2

=

2

cos

.

Annette Pilkington

Trigonometric Substitution

Trigonometric Substitution Integrals involving

a2 - x2

Integrals

involving

p x

2

+

a2

Integrals involving

Integrals involving a2 - x2

x2 - a2

We make

a2 - x2

the substitution

=

p a2

-

a2

sin2

x

= =

a sin , a| cos |

-

2

=a

2

cos (since

, dx = a cos d,

-

2

2

by choice. )

Example

Z x3

dx

4 - x2

Let x = 2 sin , dx = 2 cos d,

4

-

x2

=

p 4

-

4

sin2

=

2

cos

.

R

x 3 dx 4-x 2

=R

8 sin3 (2 cos d) 2 cos

=R

8 sin3 d = R

8 sin2 sin d =

8 R (1 - cos2 ) sin d.

Annette Pilkington

Trigonometric Substitution

Trigonometric Substitution Integrals involving

a2 - x2

Integrals

involving

p x

2

+

a2

Integrals involving

Integrals involving a2 - x2

x2 - a2

We make

a2 - x2

the substitution

=

p a2

-

a2

sin2

x

= =

a sin , a| cos |

-

2

=a

2

cos (since

, dx = a cos d,

-

2

2

by choice. )

Example

Z x3

dx

4 - x2

Let x = 2 sin , dx = 2 cos d,

4

-

x2

=

p 4

-

4

sin2

=

2

cos

.

R

x 3 dx 4-x 2

=R

8 sin3 (2 cos d) 2 cos

=R

8 sin3 d = R

8 sin2 sin d =

8 R (1 - cos2 ) sin d.

Let w = cos , dw = - sin d,

Z 8

(1-cos2 ) sin

d

=

Z -8

(1-w 2)

dw

Z =8

(w 2-1) dw

=

8w 3 -8w +C

3

=

8(cos )3 3

- 8 cos + C

=

8(cos(sin-1 3

x 2

))3

- 8 cos(sin-1

x 2

)

+

C

.

Annette Pilkington

Trigonometric Substitution

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