FURTHER INTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)

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1 Mathematics Revisio Guides More Trigoometric ad Log Itegrals Page of 7 MK HOME TUITION Mathematics Revisio Guides Level: AS / A Level AQA : C Edexcel: C OCR: C OCR MEI: C FURTHER INTEGRATION TECHNIQUES TRIG, LOG, EXP FUNCTIONS) Versio : Date: 7-- Example is copyrighted to its ower ad used with their permissio

2 Mathematics Revisio Guides More Trigoometric ad Log Itegrals Page of 7 Further Itegratio Techiques Trigoometric Itegrals We have already met several trigoometric itegrals i earlier sectios The followig two are typical examples / Example ): Fid x cos x / 6 Remember radia measure must be used!) We ca use the tabled result xcos x ad obtai the aswer x+ c x+ c x c Alteratively, we ca look at the itegral ad otice that it icludes a power of x the fourth power) multiplied by its derivative a chai rule result We ca thus guess that the itegral will be somethig like x compare itegratig x to get x ) Differetiatig x gives x cos x, but our origial itegral was x cos x The guess is too large by a factor of, so we eed to multiply it by to brig it to scale This gives x+ c as before / This is a defiite itegral, so its value is Remember: / = ; /6 = ½) x ) / 6 Example ): Fid x ta We ca use the tabled result sec ad obtai the aswer sec x+ c sec x x ta x sec x c Alteratively, we ca rewrite the itegrad as sec x sec x ta x, thus showig the product of the cube of sec x ad its derivative, sec x ta x, more clearly This suggests a aswer of the form sec x Differetiatig sec x gives sec x sec x ta x sec x ta x This result is too large by a factor of, therefore the true itegral is sec x+ c as above

3 Mathematics Revisio Guides More Trigoometric ad Log Itegrals Page of 7 There are also may other trigoometric itegrals which ca be evaluated by ug idetities ad compoud agle formulae to simplify complicated itegrals ito forms which are easier to itegrate The double agle formulae for cos A are especially useful cos A+ A= This is the Pythagorea idetity) + ta A sec A cot Acosec A A) = A cos A cos A) = cos A - A = cos A = - A ta A) = ta A -ta A From the formula for cos A) cos A = ½ + cos A) A = ½ - cos A) The triple agle formulae for cos A) ad A) also crop up at times: cos A = cos A cos A A = A - A cos A cos A cosa) A A A) Examiatio questios o trigoometric itegrals of this type geerally have a itroductory hit as to the correct method to be used, ad are ot quite as difficult as some of the examples here Example ): Fid x cosx This itegrad is based o the double-agle formula for A: A = A cos A Let A = x, ad the itegrad becomes 6x evaluatig to cos 6x c The workig is as i previous examples) Iterestigly, we could have guessed a itegral of the form of x) ad differetiated it to give 6 x) cos x), ad the scaled the fial result to obtai 6 x) + c Those seemigly differet results make sese because 6 x) = cos 6x, i other words, they differ by the costat cos x x Example ): Fid This time we have the formula cos A = cos A A i disguise Let A = x, ad the itegrad becomes simply cos x or x + c Example ): Fid / ta Here we use the idetity ta x = sec x, to get x Leave the result i surds ad terms of / / This is a stadard itegral : the result is Remember: ta /) = ) sec x ta x x

4 Mathematics Revisio Guides More Trigoometric ad Log Itegrals Page of 7 Itegratio of powers of x ad cos x These itegrals crop up quite frequetly, but they come i two types: Eve powers of x ad cos x oly If the expressio is a eve power of x or cos x or a product of the two), the techique is to rewrite the itegral as a product of terms i cos x ad /or x From there, we ca use the formulae for cos x to replace occurreces of cos x with ½ + cos x), ad occurreces of x with ½ - cos x) These forms are easier to itegrate Example 6): Fid cos x The itegrad simplifies ito ½ + cos x), givig a result of x x Example 7): Fid x x c / x Rewrite the itegrad as x = cos x ) cosx) cos x This expads to )), but there is still a ier term i cos x) which eeds simplifyig: amely cos x) = ½ + cos x) The fial expasio of the itegrad therefore gives cosx) cos x )) or cosx) cos x )) / This ca ow be itegrated to give x x) x )) / x x x = Remember: / = ; = ) Example): Fid x cos x Rewrite the itegrad as cos x) cos x ) = This simplifies ito the differece of squares form of cos x)) cosx )) x cos x The resultig itegrad ca be simplified agai to )) ) Itegratio gives x x) c = x x) c Example9): Fid cos x) Rewrite the itegrad as cos x) cosx )) cosx) cos x This expads to )), ad the we replace cos x) = ½ + cos x) The fial expasio of the itegrad therefore gives cosx) cos x )) cosx) cos x )) This ca ow be itegrated to give x x) x x )) x x) 6x) c

5 Mathematics Revisio Guides More Trigoometric ad Log Itegrals Page of 7 At least oe power of x or cos x is odd If the expressio has at least oe odd power of x ad/or cos x, we use the idetity cos x + x= to covert the expressio ito a form which cotais terms of the form x cos x ad/or cos x x These forms are reverse chai rule results which itegrate to + x + c ad cos + x + c Example ): Fid / x Rewrite the itegrad as x x, ad thus as -cos x) x Expasio by the biomial theorem gives - cos x + cos x) x The itegrad the becomes a sum of reversed chai rule results: / x cos x x cos x x / This itegrates to cos x cos x cos x ) ) Note cos /) =, cos = ) Some itegrals ca be evaluated ug alterative methods: Example ): Fid / cos x Method : Rewrite the itegrad as cos x cos x, ad thus as - x) cos x The itegrad thus becomes: / cos x xcos x / Itegratio gives x x Method : Use the triple agle formula: / cos / ) x = cos cosx Note /) =, /) = -) Example ): Fid cos x x Note /) = ) x = / x x Rewrite the itegrad as as x - 6 x) cos x x cos x xcos x which itegrates to 7 x x c 7,

6 Mathematics Revisio Guides More Trigoometric ad Log Itegrals Page 6 of 7 Other applicatios of logarithmic itegrads Some trigoometric itegrads ca also led themselves to a logarithmic fuctio whe itegrated Examples ) : Ug the fact that the derivative of sec x is sec x ta x or otherwise, fid i) ta x ad use the result to ii) evaluate / ta x I i) we ca rewrite the idetity derivative of the bottom lie sec x ta x ta x, givig us a itegrad where the top lie is the sec x ta x = l sec x + c Alteratively, we could have used the idetity where the top lie is the derivative of the bottom lie ta x = - l cos x + c x) ta x cos x, agai givig us a itegrad The two itegrals are equivalet ce cos x ad sec x are reciprocals of oe aother For part ii) we rewrite the itegrad as sec x )ta x) or ta x sec x ta x Ug the reverse chai rule as i Trigoometric Itegrals ) ad the result from part i) of the questio, / we have the itegral ta lsec x ) x or l There is o eed to iclude the modulus sig aroud the logarithm, ce sec x > for x i the rage Note ta /) = ; sec /) = ; sec) = With some other ratioal itegrads, the top lie might ot be exactly the derivative of the bottom lie, but we ca use iverse trig fuctios ad algebraic adjustmet / Example ): Fid the value of x x Hit: d ta x) x The derivative of the deomiator is x, but the umerator is ot quite right for the itegral to be a straightforward logarithmic result We ca however rewrite the itegrad as x x x The first term of the itegrad is ow a valid logarithmic fuctio ad the secod term a iverse trig fuctio, givig a itegral of l x ) ta x = l ) l) l There is o eed to iclude the modulus sig aroud the logarithm, ce x + > for all x Remember ta - ) = /)

7 Mathematics Revisio Guides More Trigoometric ad Log Itegrals Page 7 of 7 Sometimes a quadratic deomiator ca be factorised ad the itegrad rewritte i partial fractios Examples ): Fid the value of : i) iii) 6 x x x x 9 x 7 ; ii) x x 6 x x, givig the result as a gle logarithm Noe of the itegrads have the top lie equal to a multiple of the derivative of the bottom lie, but each ca be re-expressed i partial fractios I i) the deomiator ca be factorised to x-)x-) ad the itegrad rewritte as x x Full workig is i Example ) of the documet Partial Fractios ) Copyright OUP, Uderstadig Pure Mathematics, Sadler & Thorig, ISBN 97999, Exercise A, Q ) ; The itegral is therefore l x- ) + l x- ) + c This same result ca also be expressed as l x ) x ) c or l A x ) x ) where A is o-zero I ii) the deomiator ca be factorised to x-)x+) ad the itegrad rewritte as x x Full workig is i the opeig paragraph of the documet Partial Fractios ) The itegral is therefore l x- ) - l x+ ) + c This same result ca also be expressed as x ) l x ) c or x ) l x ) A where A > I iii) the deomiator ca be factorised to x - )x+) ad the itegrad rewritte as 6 x x Full workig is i Example ) of the documet Partial Fractios ) l x ) l x ) The itegral is therefore 6 = l x ) x ) 6 l 7 l = l 9