Lesson_10

# Lesson_10 - 'n1 ilxbhpi`e il`ivpxtic oeayg (104010) 10...

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Unformatted text preview: 'n1 ilxbhpi`e il`ivpxtic oeayg (104010) 10 lebxz daiib ixei :dkixr oxia xnr ,cwy inrp x&amp;quot;c ,ipinipa a`ei 'text :ddbd uixw di`n :dqtcd mieqn `l lxbhpi`-y jk F ( x ) divwpet `id I rhwa f ( x ) divwpetd ly (mieqn `l lxbhpi` e`) dnecw divwpet f` , F ( x ) = f ( x ) m` okle reawa efn ef zelcap zenecw zeivwpet izy lk . F ( x ) = f ( x ) ly ziqiqa dniyx mixikn ep` . Z f ( x ) d x = F ( x ) + C i&amp;quot;r zpzip zillkd dnecwd divwpetd 1 .zeiqiqad zeivwpetd ly zexfbpd zniyx jetidn milawznd - &amp;quot;miiciin milxbhpi`&amp;quot; mi`ad miiqiqad millkd ipy z` miaipn ( f + g ) = f + g- e ( af ) = af dxifbd illk ipy :milxbhpi`l Z ( f ( x ) + g ( x ) ) d x = Z f ( x ) d x + Z g ( x ) d x Z αf ( x ) d x = α Z f ( x ) d x 1 libxz :mi`ad milxbhpi`d z` eayg Z (3 x- 1) 2 x d x (`) Z tan 2 x d x (a) Z d x 3 + 2 x (b) reaw a &amp;gt; , Z d x x 2 + a 2 (c) Z d x 1 + cos x (d) ly 3 cenra ,qxewd ly (oyid) xz`ay &amp;quot;invr cenill xneg - miieqn-`l lxbhpi`&amp;quot; uaewa `evnl ozip dniyxd z` 1 .14 -e 11, 10 ,9 ,8 ,7 ,6 ,3 ,2 ,1 :md ef dniyxa mdixtqny milxbhpi`d z` miiciinl aiygp .uaewd 1 :oexzt (`) Z (3 x- 1) 2 x d x = Z 9 x 2- 6 x + 1 x d x = 9 Z x d x- 6 Z d x + Z 1 x d x = 9 2 x 2- 6 x +ln | x | + C (a) Z tan 2 x d x = Z 1 cos 2 x- 1 d x = tan x- x + C :zeletk zeiefl zeiedfd oke ,ef zedf . tan 2 x = 1 cos 2 x- 1 zedfa epynzyd) sin 2 x = 2 sin x cos x cos 2 x = cos 2 x- sin 2 x = 2 cos 2 x- 1 = 1- 2 sin 2 x (.t&amp;quot;ra oze` xekfl yie ,jynda d`xpy itk ,milxbhpi` iaeyiga daxd zeriten ln | 3 + 2 x | zxifbay xg`n :dwicae yegipa xfrdl ozip df sirqa lxbhpi`d aeyigl (b)-y wiqdle mi`zn reawa ltk ici-lr &amp;quot;owz&amp;quot;l ozip , 2 3 + 2 x milawn Z d x 3 + 2 x = 1 2 ln | 3 + 2 x | + C-y wecale ygpl lw illk ote`a Z f ( ax + b ) d x = 1 a F ( ax + b ) + C f` , Z f ( x ) d x = F ( x ) + C m` .mireaw b-e a 6 = 0 lkl-y lawp ,(b) sirqay oexwrd it-lr (c) Z d x x 2 + a 2 = 1 a 2 Z d x ( x a ) 2 + 1 = 1 a 2 · a arctan x a = 1 a arctan x a + C Z d x 1 + cos x = Z d x 2 cos 2 x 2 = tan x 2 + C (d) miwlga divxbhpi` :ozep ( uv ) = u v + uv dgqepd ly jetid 2 miwlga divxbhpi` zgqep Z u ( x ) v ( x ) d x = u ( x ) v ( x )- Z u ( x ) v ( x ) d x .(epnn wlg e`) cpxbhpi`d z` ,mvnvl e` ,hytl `ed miwlga divxbhpi` ly ipiit` yeniy oeni` zyxec `ide dil`n zpaen dpi` u,v ly dxigad oldly (b) e` (a) mitirqa enk ,minrtl .oeiqpe 2 libxz :mi`ad milxbhpi`d z` eayg Z x 2 sin x d x (`) Z arctan x d x (a) Z x 2 ( x 2 + 1) 2 d x (b) :oexzt gwip (`) u ( x ) = x 2 v ( x ) = sin x u ( x ) = 2 x v ( x ) =- cos x f`e Z x 2 sin x d x =- x 2 cos x + Z 2 x cos x d x =- x 2 cos x + 2 Z x cos x d x :gwip .miwlga divxbhpi`a aey ynzyp dfd lxbhpi`d aeyigl u ( x ) = x v ( x ) = cos x u ( x ) = 1 v ( x ) = sin x f`e Z x cos x d x = x sin x- Z sin x d x = x sin x + cos x + C :lawp mcewd oeieya davd ici-lr Z x 2 sin x d x =- x 2 cos x + 2 x sin x + 2 cos x + C ick , v-e u zxigaa miiawr zeidl aeyg miwlga divxbhpi`a xfeg yeniya :dxrd .`vend dcewpl xefgl `l gwip (a) u ( x ) = arctan x v ( x ) = 1 u ( x ) = 1 1 + x 2 v ( x ) = x...
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## This note was uploaded on 03/12/2011 for the course MATH 104010 taught by Professor Dr.miriambarazinna during the Winter '11 term at Technion.

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Lesson_10 - 'n1 ilxbhpi`e il`ivpxtic oeayg (104010) 10...

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