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1 C Integration - By sbstittion PhysicsAndMathsTtor.com. Using the sbstittion cos +, or otherwise, show that e cos + sin d e(e ) (Total marks). (a) Using the sbstittion cos, or otherwise, find the eact vale of d ( ) (7) The diagram above shows a sketch of part of the crve with eqation y, ( ) < <. The shaded region S, shown in the diagram above, is bonded by the crve, the -ais and the lines with eqations and. The shaded region S is rotated throgh radians abot the -ais to form a solid of revoltion. (b) Using yor answer to part (a), find the eact volme of the solid of revoltion formed. () (Total marks) Edecel Internal Review

2 C Integration - By sbstittion PhysicsAndMathsTtor.com. (a) Find tan d. () (b) Use integration by parts to find ln d. () (c) Use the sbstittion + e to show that e + e d e e + ln( + e ) + k, where k is a constant. (7) (Total marks). Use the sbstittion to find the eact vale of ( + ) d. (Total marks) 5. Using the sbstittion, or otherwise, find the eact vale of 5 d ( ) (Total 8 marks) Edecel Internal Review

3 C Integration - By sbstittion PhysicsAndMathsTtor.com. Use the sbstittion sin θ to find the eact vale of ( ) d. (Total 7 marks) 7. Use the sbstittion + sin and integration to show that sin cos ( + sin ) 5 d ( + sin ) [ sin ] + constant. (Total 8 marks) 8. Use the sbstittion ( ) to find ( ) d, giving yor answer in terms of. (Total marks) 9. Use the sbstittion + to find the eact vale of ( + ) d. (Total marks) Edecel Internal Review

4 C Integration - By sbstittion PhysicsAndMathsTtor.com. sin d B cos sin e + d e M A e ft sign error A ft cos e + cos + ( ) e e e or eqivalent with M e( e ) cso A []. (a) d sin B d sin M cos cos ( ) ( ) sin Use of cos sin M cos sin cos d ± k cos M tan ( + C) ±k tan M cos cos M tan tan tan ( ) A 7 Edecel Internal Review

5 C Integration - By sbstittion PhysicsAndMathsTtor.com (b) V d ( ) M d integral in (a) M their answer to part (a) Aft []. (a) tan d [NB : sec A + tan A gives tan A sec A ] The correct nderlined identity. sec d M oe tan (+c) Correct integration with/withot + c A (b) n d n d dv v d n. d Use of integration by parts formla in the correct direction. M Correct direction means that ln. Correct epression. A n + d An attempt to mltiply throgh k n n, n... by and an attempt to... n + ( + c)... integrate (process the reslt); M correct soltion with/withot + c A oe Edecel Internal Review 5

6 C Integration - By sbstittion PhysicsAndMathsTtor.com e (c) d + e + e e d d d,, Differentiating to find e any one of the three nderlined B e.e ( ).e d. e Attempt to sbstitte for + e e d f(), their and + e e ( ) or. or e d f(), their M * ( ) and + e. ( ) ( ) A + An attempt to mltiply ot their nmerator to give at least three terms + and divide throgh each term by dm * + n ( + c) Correct integration with/withot +c A ( + e ) ( + e ) + n(+ e ) + c Sbstittes + e back into their integrated epression with at least two dm * terms. + e + e e + n ( + e ) + c e + e + e + ln( + e ) + c e e + ln( + e ) + c e e + ln( + e ) + k AG e e + n(+ e ) + k mst se a + c + and combined. A cso 7 [] Edecel Internal Review

7 C Integration - By sbstittion PhysicsAndMathsTtor.com. d, with sbstittion ( + ) d.ln d.ln d + d ( ) ln ( + ) c ln + ( + ) B d.ln or d M* k ( + ) d.ln or ln d where k is constant M ( + ) a( + ) (*) A ( + ).( + ) (*) (*) If yo see this integration applied anywhere in a candidate s working then yo can award M, A change limits: when & then & ( + ) d ln ln ln ( + ) depm Correct se of limits and Aaef or or (*) ln ln ln 8 ln ln Eact vale only! (*) There are other acceptable answers for A. eg: NB: Use yor calclator to check eg ln 8 or ln Edecel Internal Review 7

8 C Integration - By sbstittion PhysicsAndMathsTtor.com Alternatively candidate can revert back to... d ( + ) ln ( ) + ln ln depm* Correct se of limits and Aaef or or (*) ln ln ln 8 ln ln Eact vale only! (*) There are other acceptable answers for A. eg: NB: Use yor calclator to check eg ln 8 or ln [] 5. Uses sbstittion to obtain f() and to obtain d +, M const. or eqiv. M ( + ) Reaches or eqivalent A Simplifies integrand to + or eqiv. M Integrates to + Aft dependent on all previos Ms Uses new limits and sbstitting and sbtracting (or retrning to fnction of with old limits) M To give cso A 8 [8] Edecel Internal Review 8

9 C Integration - By sbstittion PhysicsAndMathsTtor.com By Parts Attempt at right direction by parts M [ ( ) { ( ) d}] M{MA}... ( ) MAft Uses limits 5 and correctly; [ ] MA. d ( ) ( sin θ ) cos θ dθ M d Use of sinθ and cosθ dθ θ cos θ d M A sec θ dθ tanθ M A Using the limits and to evalate integral M θ cao A [ tan ] Alternative for final M A Retrning to the variable and sing the limits and to evalate integral M ( ) cao A [7] 7. + sin d cos or cos d or d cos M I 5 ( ) M, A Fll sb. to I in terms of, correct 5 ( ) M Edecel Internal Review 9

10 C Integration - By sbstittion PhysicsAndMathsTtor.com Correct split 7 (+ c) M, A 7 M for n n+ ( 7) (+ c) M Attempt to factorise ( + sin ) ( sin + 7 ) + c ( + sin ) ( sin ) (+ c) (*) A cso [8] Alt: Integration by parts I ( ) Attempt first stage M M 7 ( ) Fll integration A 7 ( rest as scheme 7 or 7 7 ) 8. + M A A d ( + ) I ( + + ) c 5 A M A 5 ( 5 ) + ( ) + ( ) + c M A [] Edecel Internal Review

11 C Integration - By sbstittion PhysicsAndMathsTtor.com 9. Uses To give d M Integrates to give Uses correct limits and (or and for ) To obtain 8 A M, A + A M [] Edecel Internal Review

12 C Integration - By sbstittion PhysicsAndMathsTtor.com. This qestion was generally well done and, helped by the printed answer, many proced flly correct answers. The commonest error was to omit the negative sign when differentiating cos +. The order of the limits gave some difficlty. Instead of the correct incorrect version e, an e was proced and the reslting epressions maniplated to the printed reslt and working like (e e ) e + e e(e ) was not ncommon. Some candidates got into serios difficlties when, throgh incorrect algebraic maniplation, they obtained e sin instead of e. This led to epressions sch as e ( ) and the efforts to integrate this, either by parts twice or a frther sbstittion, often ran to several spplementary sheets. The time lost here inevitably led to difficlties in finishing the paper. Candidates need to have some idea of the amont of work and time appropriate to a mark qestion and, if they find themselves eceeding this, realise that they have probably made a mistake and that they wold be well advised to go on to another qestion.. Answers to part (a) were mied, althogh most candidates gained some method marks. A srprisingly large nmber of candidates failed to deal with cos correctly and many did not recognise that d sec d tan ( + C) in this contet. Nearly all cos converted the limits correctly. Answers to part (b) were also mied. Some cold not get beyond stating the formla for the volme of revoltion while others gained the first mark, by sbstitting the eqation given in part (b) into this formla, bt cold not see the connection with part (a). Candidates cold recover here and gain fll follow throgh marks in part (b) after an incorrect attempt at part (a).. In part (a), a srprisingly large nmber of candidates did not know how to integrate tan. Eaminers were confronted with some strange attempts involving either doble angle formlae or logarithmic answers sch as ln(sec ) or ln(sec ). Those candidates who realised that the needed the identity sec + tan sometimes wrote it down incorrectly. Part (b) was probably the best attempted of the three parts in the qestion. This was a tricky integration by parts qestion owing to the term of, meaning that candidates had to be especially carefl when sing negative powers. Many candidates applied the integration by parts k formla correctly and then went on to integrate an epression of the form to gain ot of the marks available. A significant nmber of candidates failed to gain the final accracy mark owing to sign errors or errors with the constants α and β in α ln + β + c. A minority of candidates applied the by parts formla in the wrong direction and incorrectly stated that d v dh ln implied v. In part (c), most candidates correctly differentiated the sbstittion to gain the first mark. A significant proportion of candidates fond the sbstittion to obtain an integral in terms of more demanding. Some candidates did not realise that e and e are (e ) and (e ) respectively and hence, rather than ( ) was a freqently encontered error seen in the Edecel Internal Review

13 C Integration - By sbstittion PhysicsAndMathsTtor.com nmerator of the sbstitted epression. Fewer than half of the candidates simplified their ( ) sbstitted epression to arrive at the correct reslt of. Some candidates cold not proceed frther at this point bt the majority of the candidates who achieved this reslt were able to mltiply ot the nmerator, divide by, integrate and sbstitte back for. At this point some candidates strggled to achieve the epression reqired. The most common misconception was that the constant of integration was a fied constant to be determined, and so many candidates conclded that k. Many candidates did not realise that when added to c combined to make another arbitrary constant k.. Many candidates had difficlties with the differentiation of the fnction, despite the same problem being posed in the Janary 7 paper, with incorrect derivatives of and d d being common. Those candidates who differentiated with respect to to obtain either ln or often failed to replace with ; or if they did this, they failed to cancel the variable from the nmerator and the denominator of their algebraic fraction. Therefore, at this point candidates proceeded to do some very complicated integration, always with no chance of a correct soltion. Those candidates who attempted to integrate k( + ) sally did this correctly, bt there were a significant nmber of candidates who either integrated this incorrectly to give k( + ) or ln f(). There were a significant proportion of candidates who proceeded to integrate ( + ) with respect to and did so by either treating the leading as a constant or sing integration by parts. Many candidates correctly changed the limits from and to and to obtain their final answer. Some candidates instead sbstitted for and sed limits of and. 5. Most candidates chose to se the given sbstittion bt the answers to this qestion were qite variable. There were many candidates who gave sccinct, neat, totally correct soltions, and generally if the first two method marks in the scheme were gained good soltions sally followed. The biggest problem, as sal, was in the treatment of d : those who differentiated implicitly were sally more sccessfl, in their sbseqent maniplation, than those who chose to write ( ) and then find ; many candidates ignored the d d altogether, or effectively treated it as, and weaker candidates often integrated an epression involving both and terms. Some candidates spoilt otherwise good soltions by applying the wrong limits.. This was a qestion which candidates tended to either get completely correct or score very few marks. If the d is ignored when sbstitting, an integral is obtained which is etremely difficlt to integrate at this level and little progress can be made. Those who knew how to deal with the d often completed correctly. A few, on obtaining tan θ, sbstitted and instead of Edecel Internal Review

14 C Integration - By sbstittion PhysicsAndMathsTtor.com and. Very few attempted to retrn to the original variable and those who did were rarely sccessfl. 7. Most candidates were able to make a start here bt a nmber did not progress mch beyond cos d. Some failed to realize that sin ( ) and others tried integrating an epression 5 with a mitre of terms in and. Those who reached ( ) sally went on to score the first marks, bt the final two marks were only scored by the most able who were able to complete the factorization and simplification clearly and accrately. 8. No Report available for this qestion. 9. No Report available for this qestion. Edecel Internal Review

Time: 1 hour 30 minutes

Time: 1 hour 30 minutes Paper Reference(s) 6666/0 Edecel GCE Core Mathematics C4 Gold Level (Harder) G Time: hour 0 minutes Materials required for eamination Mathematical Formulae (Green) Items included with question papers Nil