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.