4012 u-du : integrating composite functions ap calculus
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4012 u-du: Integrating Composite Functions
AP Calculus
Trig Functions
( ) cot( )tan x dx x dx
sec( ) csc( )x dx x dx
au
ua du
Formal Change of Variables << the Extra “x”>>
Solve for x in terms of u ILL: Let
then
and becomes
2 6 *2x x dx (2 6)u x 6
2
ux
2du dx6
* *2
uu du
Review: Antiderivatives
Formal Change of Variables << the Extra “x”>>
Rewrite in terms of u - du
4(2 1)x x dx
Formal Change of Variables << the Extra “x”>>
Rewrite in terms of u - du
2 1
3
xdx
x
Formal Change of Variables << Changing the Boundaries>>
Rewrite in terms of u - du
422
0
6x
x
x x dx
NEW :Definite Integrals
Change of Boundaries
4 9
0 1
12 1 =
2x dx u du
x
y
x
y
Formal Change of Variables << Changing the Boundaries>>
Rewrite in terms of u - du
NEW :Definite Integrals
2
21 3 5
x
x
dx
x
EX: Change of Variables
3
21
1
2
xdx
x x
ln
Formal Change of Variables << Changing the Boundaries>>
Solve for x in terms of u
7
0
2 x
x
x x dx
Complete Change of Variables << Changing du >>
At times it is required to even change the du as the u is changed above.
1cos
2x dx u x du dx
x
2
2
xdu dx
u du dx
cos 2u u du
We will solve this later in the course.
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