5044 Integration by Parts

AP Calculus

Integration by Parts• Product Rules for Integration :

• A.    Is it a function times its derivative; u-du

• B.     Is it a function times the derivative of a second

function; u-dv

Integration by Parts

( )

( )

( )

d dv duuv u v

dx dx dx

dv d duu uv v

dx dx dx

dv d duu dx uv dx v dx

dx dx dx

udv uv vdu

udv uv vdu

Proof with u-v :worked out

( )d dv du

uv u vdx dx dx

( )dv d du

u uv vdx dx dx

( )dv d du

u dx uv dx v dxdx dx dx

( ) ' ' 'uv uv vu

udv uv vdu Proof with u-v

Proof / Formula• Integration by parts

( )* ( ) ( )* '( ) ( )* '( )d

f x g x f x g x g x f xdx

• In words:• First * Second minus of Second * D (first )

Integration by parts : worked out

( )* ( ) ( )* '( ) ( )* '( )d

f x g x f x g x g x f xdx

( )* '( ) ( )* ( ) ( )* '( )f x g x dx f x g x g x f x dx

( )* '( ) ( )* ( ) ( )* '( )d

f x g x dx f x g x dx g x f x dxdx

( )* '( ) ( )* ( ) ( )* '( )d

f x g x f x g x g x f xdx

In words:First * Second minus of Second * D (first )

Illustration:xxe dx

Choosing u and dv L I P E T L Logarithmic

I Inverse Trig

P Polynomial

E Exponential

T Trig

Let u be the first function from the mnemonic.

Illustration:xxe dx u = dv =

du = v =

Examples:

2 ln( )x x dx

Examples: (hidden product)

arcsin( )x dx

Examples: ( hidden product - ln )

ln( )x dx

LEARN IT ! , LOVE IT !, LIVE IT ! Write it on your Anti-derivative page.

Chart: Short Cut for Algebraic

1 1

2 2

3 3

0

u dv

du v

du v

du v

+

_

+

_

Examples: (shortcut for algebraic u)4 xx e dx

Examples: (shortcut for algebraic u)3 cos( )x x dx

Examples: (Solve like an equation)

sin( )xe x dx

Examples: (Solve like an equation)

2 cos(3 )xe x dx

S. O. B. Problem

Creativity is ALWAYS the Main Ingredient!

3sec ( )x dx

Last Update:

• 03/08/12

• Assignment: p. 346 # 1 – 29 odd

sin(3 ) s( )x co x dx