5014-integration inverse trig
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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
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