5014-Integration Inverse Trig

AP Calculus

Remember Patterns:

1sin ( )y xy

x1

2(1 )x

1tan ( )y xy

x

1

2( 1)x

1sec ( )y xy

x

1

2( 1)x

Remember Derivaties:

1

2

1sin ( )

(1 )

du u

dx u

12

1tan ( )

1

du u

dx u

1

2

1sec

( 1)

du u

dx u u

Generalize:

29

dx

x Looks like arcsin pattern except NOT 1 -

MEMORIZE !

2 2arcsin

du uc

aa u

2 2

1arctan

du uc

a u a a

2 2

1sec

uduarc ca au u a

a is a constant. u = f (x)

Examples:

24

dx

x Straight Plug In

Examples:

22 9

dx

x Normal U - du Problem ?

Inverse Trig.

(also has a U-du component)

Arctan vs. Ln

27 4

dx

x27 4

xdx

x

Always check ln first !

Example (introduce a variable factor):

4

9 4

dx

x x arcsec ????

U - du ????

REM: NEVER CROSS the with a variable ! ! !

Example (introduce a variable factor):

2

1x

dx

e arcsec ???? U - du

Examples (Complete the square):

2 8 21

dx

x x

Examples (Complete the square):

26

dx

x x Negative lead

Examples (Complete the square):

22 8 10

dx

x x Non-one lead

Examples (THINK):

2

2

6 13

xdx

x x ADD ZERO

Last Update:

• 02/06/06

• Assignment: Worksheet Larson 5.8