Math I B ( 8:30AM ) 4 Feb 2019

Using Integration by parts to derive"

reduction formulas.

"

Derive a reduction formula for ) cos"

X DX ( where h

is an integer ,n -327

.

I .cn. "

" " " d " cosh- 'xsinxtaiiscoixhiiixdxAn

I cos" Xdx -

- cos" - '

xsinxi - A - Df cos "x ( I - cos.

X ) dx

Ifcos

" xdx -

- cosh- '

xsinxt A - tf cosh-

Xd X - G - DJ cos"

X dx+ Call Scoixdx t Cn - D Scoshxdx

hfcoixdx -

- cos" - '

xsinxt Ca - Ascot - ' xdx

Reduction

: /Sco5Xdx=tCos"xsinxth÷fcos"xd#We have reduced the powerof the cos in the integral uh

from n to n - 2. 7 Scosoxdx -

- X t C

) cos 'Xdx = sin Xtc

> S cos' xdx -

- I cosxsinxt 'T Scosoxdx= I cosxsinxt I X t C

I cos 'xdx -

- T coixsinxt F fcoixdx= 'T Cos

'

xsinxt F sing t C

Icoixdx = IT Cos 'X sinxt ITS coixdx= 4 cos

'

Xsinxt ¥ ⇐ cosxsinxt TX ] t C

= 4 cos'

x sinxt I cosxsinxt TX

We mainly need this formula to integrate Ten

powers of cos X ,We 'll find an easier way to do

Odd powers of cos X soon .

Integrating Isin "

X cosmxdx where n,

on are integerssincey5

is odd,

we can peel off one factor

CAsE## m is odd of cos X, leaving an . even

example : Ssinxdxcosxdx power of cos X

even o ↳= Isin " xcoixcosxdx u -

- six du -- oosxdx

- Spade

=

.

Isin "

x G- sink)"

cos xdx

-- Sui Ci - ay

'

du= f ( us - 2h 't u

" ) du= fu '

- Fu 't FYI c = at sin"

x - Tsin?Xttssinsxtc

CASE # 2 : h is odd strip off a sinxdx

example I sink cos 'xdx to be our- da

- u -

. cosxdxLeaving an even power

= S sin'

X cos 'xsinxdx da.

- sinxdx

of sin x , allowing us to - da = sinxdx

conversing = > I - cosy= SCI - Cos 'x ) coixsinxdx

a" Osx

= - f a - uyu-

du -

- f ( - u'

tu 4) du -

- tu'

UIC

=- T cos

'

Xt 'T Cos C

CASE # 3 CThe hard one ) : n and m are both even

example : Isin "x coixdhx

In this case we use

the Pythagorean identity= flint )

-

cos'

xdxto turn

everything into cosines

( or sines and use the reduction" I ( I - cos 'xT Cos

'

xdxformula .

= f cog'

xdx - Szcoixdxt Scoixdx

Tl 'TUse reduction formula