Integration using a table of anti derivatives

Transcription

1 Itegrtio usig tble of ti derivtives mc-stck-ty-ittble-009- We my regrd itegrtio s the reverse of differetitio. So if we hve tble of derivtives, wecreditbckwrdsstbleofti-derivtives.whewedothis,weofteeedtodel with costts which rise i the process of differetitio. I order to mster the techiques explied here it is vitl tht you udertke plety of prctice exercises so tht they become secod ture. Afterredigthistext,d/orviewigthevideotutorilothistopic,youshouldbebleto: usetbleofderivtives,ortbleofti-derivtives,iordertoitegrtesimplefuctios. Cotets. Itroductio. Itegrtig powers. Itegrtig expoetils. Itegrtig trigometric fuctios 5. Itegrls givig rise to iverse trigoometric fuctios 5 c mthcetre 009

2 . Itroductio Whe we re itegrtig, we eed to be ble to recogise stdrd forms. The followig tble gives list of stdrd forms, obtied s ti-derivtives. Sometimes, it my be possible to use oe of these stdrd forms directly. O other occsios, some mipultio will be eeded first. Key Poit x dx = x+ + c ( + (x + b (x + b+ dx = + c ( ( + dx = l x + c x x + b dx = l x + b + c e x dx = e x + c e mx dx = m emx + c cos xdx = si x + c cos xdx = six + c si xdx = cos x + c si xdx = cos x + c sec xdx = t x + c sec xdx = t x + c dx = x si x + c ( x dx = x si + c + x dx = t x + c + x dx = t + c c mthcetre 009

3 . Itegrtig powers Wekowthtthederivtiveof x is x.replcig by + weseethtthederivtiveof x + is ( + x,sothtthederivtiveof x+ + is x (providedtht + 0.Thus x dx = x+ + + c. Similrly,thederivtiveof (x + b is (x + b. Replcig by + weseethtthe derivtiveof (x + b + is ( + (x + b (x + b+,sothtthederivtiveof is (x + b ( + (providedtht + 0dtht 0.Thus (x + b (x + b+ dx = + c. ( + Whthppesif =,sotht + = 0? Wekowthtthederivtiveof l x is /x,so tht dx = l x + c. x Simlrly,thederivtiveof l x + b is x + b,sothtthederivtiveof l x + b is x + b. Thus Exmple Fid x dx. Here, = d b =,so x + b dx = l x + b + c. x dx = l x + c.. Itegrtig expoetils Wekowthtthederivtiveofe x remisuchged,se x.thus e x dx =e x + c. Similrly,wekowthtthederivtiveofe mx is me mx,sothtthederivtiveof Thus e mx dx = m emx + c. m emx ise mx. c mthcetre 009

4 Exmple Fid e x dx. Here, m =,so e x dx = ex + c.. Itegrtig trigoometric fuctios Wekowthtthederivtiveof si xis cos x.thus cosxdx = si x + c. Similrly,wekowthtthederivtiveof si xis cosx,sothtthederivtiveof si xis cos x. Thus cosx dx = si x + c. Welsokowthtthederivtiveof cosxis si x.thus si x dx = cos x + c. Similrly,wekowthtthederivtiveof cosxis si x,sothtthederivtiveof cosx is si x.thus si x dx = cosx + c. Wecusethefcttht t x = si x tofidti-derivtiveof t x. Weusetherulefor cosx logrithmicdifferetitiotoseethtthederivtiveof l cosx is si x si x t x dx = cos x dx = l cosx + c = l sec x + c. cosx,sotht (Ithelststepofthisrgumet,wehveusedthefcttht l uisequlto l(/u. There is oe more trigoometric fuctio which we c itegrte without difficulty. We kow thtthederivtiveof t xis sec x.thus sec x dx = tx + c. Similrly,thederivtiveof txis sec x,sothtthederivtiveof t xis sec x.thus sec x dx = t x + c c mthcetre 009

5 5. Itegrls givig rise to iverse trigoometric fuctios Sometimes, itegrls ivolvig frctios d squre roots give rise to iverse trigoometric fuctios. Wekowthtthederivtiveof si xis.thus x x dx = si x + c. Similrly,wekowthtthederivtiveof si is Exmple Fid dx. x x dx = si + c.,whichequls x.thus Here, = =,sotht dx = x si + c. Exmple Fid dx. 9x Thisisotquiteiourstdrdform.However,wectkethe 9outsidethesqureroot,so thtitbecomes.weget 9x dx = dx, 9 x dthisisithestdrdform.soowwectkethe outsidetheitegrl,dweseetht = =,sotht 9 dx = 9x dx 9 x = ( x si + c = ( x si + c. 5 c mthcetre 009

6 Aother type of itegrl which my be foud usig iverse trigoometric fuctio ivolves frctio, but does ot ivolve squre root. Wekowthtthederivtiveof t xis + x.thus + x dx = t x + c. Similrly,wekowthtthederivtiveof t thtthederivtiveof t is + x.thus Exmple Fid 9 + x dx. Here, = 9 =,sotht Exmple Fid 5 + 6x dx. is + x dx = t ( + ( x,whichequls + x,so + c 9 + x dx = t + c. Here,wetkethe 6outsidetheitegrl,sothtweget 5 + 6x dx = 5 dx. 6 + x 6 5 Nowwecseetht = = 5,sotht 6 ( 5 + 6x dx = 6 ( 5 t x ( 5 + c = 6 ( x 5 t + c 5 = ( x 0 t + c. 5 Exercises. Determie the itegrl of ech of the followig fuctios 8 (f sec x (b (g (c x (h 6 + x x x (d (i x 6 9x (e si 5x (j + 5x 6 c mthcetre 009

7 . Itegrtio hs the sme lierity rules s differetitio, mely kf(x dx = k f(x dx d f(x + g(x dx = Use these rules to determie the itegrls of the followig fuctios f(x dx + g(x dx. ( 5x + 0 cosx (b e x + x (c 6 sec x + e x (d x 6 si x (e x x (f 0 cos5x 5 cos 0x Aswers. I ll swers the costt of itegrtio hs bee omitted. ( 9 x9 (b x = (c x / = x (d l x x (e 5 cos 5x (f tx (g t (h si ( ( x 5x (i si (j 0 t. I ll swers the costt of itegrtio hs bee omitted si x (b e x + 8 x/ (c 9 tx e x (d 7 x7 + cos x (e 8 t + 5 si (f si 5x si 0x Lots of miscelleous dditiol exercises to eble you to prctice itegrtio usig tble re vilble o-lie courtesy of Dr Chris Sgwi d the STACK system: rdomly geerted questios utomtic mrkig feedbck d full solutios vilble. Simplyfollow thelikbyclickig oeofthestacks below: More questios like this oe: More questios like this oe: x dx: x 7 dx: 7 c mthcetre 009