Integration using trig identities or a trig substitution

Transcription

1 Integration using trig identities or a trig substitution mc-ty-intusingtrig-9- Some integrals involving trigonometric functions can be evaluated by using the trigonometric identities. These allow the integrand to be written in an alternative form which may be more amenable to integration. On occasions a trigonometric substitution will enable an integral to be evaluated. Bothofthesetopicsaredescribedinthisunit. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. Afterreadingthistext,and/orviewingthevideotutorialonthistopic,youshouldbeableto: usetrigonometricidentitiestointegrate sin x, cos x,andfunctionsoftheform sin 3x cos x. integrate products of sines and cosines using a mixture of trigonometric identities and integration by substitution use trigonometric substitutions to evaluate integrals Contents. Introduction. Integrals requiring the use of trigonometric identities 3. Integrals involving products of sines and cosines 3. Integrals which make use of a trigonometric substitution 5 c mathcentre 9

2 . Introduction Bynowyoushouldbewellawareoftheimportantresultsthat coskx dx sin kx + c sin kx dx k k coskx + c However, a little more care is needed when we wish to integrate more complicated trigonometric functions such as sin x dx, sin 3x cos x dx,andsoon. Incaselikethesetrigonometric identities can be used to write the integrand in an alternative form which can be integrated more readily. Sometimes, use of a trigonometric substitution enables an integral to be found. Such substitutions are described in Section.. Integrals requiring the use of trigonometric identities The trigonometric identities we shall use in this section, or which are required to complete the Exercises, are summarised here: sin A cosb sin(a + B) + sin(a B) cosacosb cos(a B) + cos(a + B) sin A sin B cos(a B) cos(a + B) sin A + cos A cos A cos A sin A cos A sin A sin A sin A cosa + tan A sec A Some commonly needed trigonometric identities π sin x dx. The strategy is to use a trigonometric identity to rewrite the integrand in an alternative form whichdoesnotincludepowersof sin x. Thetrigonometricidentityweshallusehereisoneof the double angle formulae: cos A sin A By rearranging this we can write sin A ( cos A) Noticethatbyusingthisidentitywecanconvertanexpressioninvolving sin Aintoonewhich has no powers in. Therefore, our integral can be written π sin x dx π ( cos x)dx c mathcentre 9

3 and this can be evaluated as follows: π ( cos x)dx [ ( x )] π sin x [ x ] π sin x π sin 3x cos x dx. Notethattheintegrandisaproductofthefunctions sin 3xand cos x.wecanusetheidentity sin A cosb sin(a+b)+sin(a B)toexpresstheintegrandasthesumoftwosinefunctions. With A 3xand B xwehave sin 3x cos x dx (sin 5x + sin x)dx ( 5 ) cos 5x cosx + c cos 5x cosx + c Exercises Use the trigonometric identities stated on page to find the following integrals. π/.(a) cos x dx (b) cos x dx (c) sin x cos x dx.(a) π/3 π/6 cos5x cos 3x dx (b) (sin t + cos t)dt (c) sin 7t sin t dt. 3. Integrals involving products of sines and cosines Inthissectionwelookatintegralsoftheform howtodealwithintegralsinwhich misodd. sin 3 x cos x dx. sin m x cos n x dx. Inthefirstexamplewesee Study of the integrand, and the table of identities shows that there is no obvious identity which willhelpushere.howeverwhatwewilldoisrewritetheterm sin 3 xas sin x sin x,andusethe identity sin x cos x.thereasonfordoingthiswillbecomeapparent. sin 3 x cos x dx (sin x sin x) cos x dx sin x( cos x) cos x dx 3 c mathcentre 9

4 Atthisstagethesubstitution u cosx, du sin x dxenablesustorapidlycompletethe solution: Wefind sin x( cos x) cos x dx ( u )u du (u u )du u5 5 u3 3 + c 5 cos5 x 3 cos3 x + c Inthecasewhen misevenand nisoddwecanproceedinasimilarfashion,usetheidentity cos A sin Aandthesubstitution u sin x. Tofind this becomes sin x cos 3 x dxwewrite sin x(cos x cosx) dx Thenthesubstitution u sin x, du cosxdxgives sin x(cos x cosx) dx.usingtheidentity cos x sin x (u u 6 )du u5 5 u7 7 + c sin5 x 5 sin x( sin x) cosxdx (sin x cosx sin 6 x cos x) dx sin7 x 7 Inthecasewhenboth mand nareevenyoushouldtryusingthedoubleangleformulae,asin Exercise Q below. + c Exercises.(a)Find cos 3 x dx (b) cos 5 x dx (c) sin 5 x cos x dx..evaluate sin x cos xdxbyusingthedoubleangleformulae sin x cos x 3. Using the double angle formulae twice find cos x sin x cos x dx. + cos x c mathcentre 9

5 . Integrals which make use of a trigonometric substitution There are several integrals which can be found by making a trigonometric substitution. Consider the following example. + x dx. Letusseewhathappenswhenwemakethesubstitution x tanθ. Ourreasonfordoingthisisthattheintegrandwilltheninvolve ( + tan A sec A)whichwillenableustosimplifythis. With x tanθ, dx dθ sec θ,sothat dx sec θ dθ.theintegralbecomes + x dx + tan θ sec θ dθ sec θ sec θ dθ dθ θ + c tan x + c So + x dx tan x + c.thisisanimportantstandardresult. We can generalise this result to the integral a + x dx: Wemakethesubstitution x a tanθ, dx a sec θ dθ.theintegralbecomes a + a tan θ a sec θ dθ andusingtheidentity + tan θ sec θthisreducesto dθ a a θ + c a tan x a + c + tan θ andwehaveanidentity Thisisastandardresultwhichyoushouldbeawareofandbepreparedtolookupwhennecessary. + x dx tan x + c Key Point a + x dx x a tan a + c 5 c mathcentre 9

6 Suppose we seek + 9x dx. Weproceedbyfirstextractingafactoroffromthedenominator: + 9x dx + 9 dx x Thisisveryclosetothestandardresultinthepreviouskeypointexceptthattheterm 9 isnot reallywanted. Letusobservetheeffectofmakingthesubstitution u 3 x,sothat u 9 x. Then du 3 dxandtheintegralbecomes + 9 dx x 6 + u 3 du + u du Thiscanbefinishedoffusingthestandardresult,togive 6 tan u + c 6 tan 3 x + c. We now consider a similar example for which a sine substitution is appropriate. a x dx. The substitution we will use here is based upon the observations that in the denominator we haveaterm a x,andthatthereisatrigonometricidentity sin A cos A(andhence (a a sin A a cos A). Wetry x a sin θ,sothat x a sin θ. Then dx a cosθand dx a cosθdθ. Theintegral dθ becomes a x dx a a sin θ a cosθdθ Hence a x dx sin x a + c. This is another standard result. a cos θ a cosθdθ a cosθ a cosθdθ dθ θ + c sin x a + c 6 c mathcentre 9

7 Key Point a x dx sin x a + c 9x dx. Thetrickistotrytowritethisinthestandardform.Let u 3x, du 3dxsothat dx 9x 3 du u Exercises 3 3 sin u + c 3 sin 3x + c. Use the trigonometric substitution indicated to find the given integral. x (a) dx let x sinθ (b) 6 x + x dx let x tan θ. Answers Exercises.(a) x + sin x + c (b) π (c) 3.(a).65(3d.p.) (b) t + c 8 (c) sin 3t sin t + c 6 Exercises cos x 8.(a) 3 cos x sin x + sin x + c (b) 3 5 cos x sin x + 5 cos x sin x + 8 sin x + c 5 (c) 7 sin x cos 3 x 35 sin x cos 3 x 8 5 cos3 x + c.. sin x cos3 x + cos x sin x + x + c c 3. 6 sin3 x cos 3 x 8 sin x cos3 x + 6 cosxsin x + 6 x + c Exercises 3.(a) x 6 x + 8 sin x + c (b) tan x + c. 7 c mathcentre 9