TERM - 2

12 CBSE

2021-22

=1D85=1D93C (As per Latest CBSE Syllabus Released in July 2021)

Jitendra Kumar MSc (Mathematics)

Full Marks Pvt Ltd (Progressive Educational Publishers) an ISO : 9001-2015 company

New Delhi-110002

Published by:

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NEW EDITION

“This book is meant for educational and learning purposes. The author of the book has taken all reasonable care to ensure that the contents of the book do not violate any existing copyright or other intellectual property rights of any person in any manner whatsoever. In the event the author has been unable to track any source and if any copyright has been inadvertently infringed, please notify the publisher in writing for corrective action.”

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Note from the Publishers MATHEMATICS-12 (TERM-2) is based on the latest curriculum released by CBSE in July 2021 for 2022 Board Examination to be conducted between March - April 2022. It will certainly prove to be a torch-bearer for those who toil hard to achieve their goal. This All-in-one Question Bank has been developed keeping in mind all the requirement of the students for Board Examinations preparations like learning, practicing, revising and assessing. SALIENT FEATURES OF THE BOOK: x Each chapter is designed in ‘Topic wise’ manner where each topic is briefly explained with sufficient Examples and Exercise. x In all chapters, Topic wise exercises cover Very Short Answers Type Questions, Short Answer Type Questions, Long Answer Type Questions - I, Long Answer Type Questions - II. MCQs, Assertion-Reasoning Type Questions and Case Study Based Questions are also included as per the Special Scheme of Assessment suggested by CBSE vide Circular No. Acad-51/2021. Answers and sufficient hints are also provided separately at the end of each exercise. x Author's Comments have been provided to suggest the students which type of questions are important for examination point of view. x Chapterwise Important Formulae and Revision Chart have been prepared for Quick Revision. x Common Errors by the students are provided to make students aware what errors are usually committed by them unknowingly. x 3 Sample Papers (2 Solved & 1 Unsolved) are given for self assessment. x The book has been well prepared to build confidence among students. Suggestions for further improvement of the book, pointing out printing errors/mistakes which might have crept in spite of all efforts, will be thankfully received and incorporated in the next edition.

(iii)

CBSE CIRCULAR 2021-22

osQUnzh; ekè;fed f'k{kk cksMZ (f'k{kk ea=kky;] Hkkjr ljdkj osQ v/hu Lok;Ùk laxBu)

CENTRAL BOARD OF SECONDARY EDUCATION (An Autonomous Organisation Under the Ministry of Education, Govt. of India)

CBSE/DIR (ACAD)/2021

Date: July 05, 2021 Circular No: A cad-51/2021

#NNVJG*GCFUQH5EJQQNUCHſNKCVGFVQ%$5' 5WDLGEV5RGEKCN5EJGOGQH#UUGUUOGPVHQT$QCTF'ZCOKPCVKQP%NCUUGU:CPF:++HQTVJG5GUUKQP COVID 19 pandemic caused almost all CBSE schools to function in a virtual mode for most part of the academic session of 2020-21. Due to the extreme risk associated with the conduct of Board examinations during the second wave in April 2021, CBSE had to cancel both its class X and XII Board examinations of the year 2021 and results are to be declared on the DCUKUQHCETGFKDNGTGNKCDNGƀGZKDNGCPFXCNKFCNVGTPCVKXGCUUGUUOGPVRQNKE[6JKUKPVWTPCNUQPGEGUUKVCVGFFGNKDGTCVKQPU over alternative ways to look at the learning objectives as well as the conduct of the Board Examinations for the academic session 2021-22 in case the situation remains unfeasible. CBSE has also held stake holder consultations with Government schools as well as private independent schools from across the country especially schools from the remote rural areas and a majority of them have requested for the rationalization QHVJGU[NNCDWUUKOKNCTVQNCUV[GCTKPXKGYQHTGFWEGFVKOGRGTOKVVGFHQTQTICPK\KPIQPNKPGENCUUGU6JG$QCTFJCUCNUQ considered the concerns regarding differential availability of electronic gadgets, connectivity and effectiveness of online teaching and other socio-economic issues specially with respect to students from economically weaker section and those TGUKFKPIKPHCTƀWPICTGCUQHVJGEQWPVT[+PCUWTXG[EQPFWEVGFD[%$5'KVYCUTGXGCNGFVJCVVJGTCVKQPCNK\GFU[NNCDWU PQVKſGFHQTVJGUGUUKQPYCUGHHGEVKXGHQTUEJQQNUKPEQXGTKPIVJGU[NNCDWUCPFJGNRGFNGCTPGTUKPCEJKGXKPINGCTPKPI objectives in a less stressful manner. In the above backdrop and in line with the Board’s continued focus on assessing stipulated learning outcomes by making the examinations competencies and core concepts based, student-centric, transparent, technology-driven, and having advance provision of alternatives for different future scenarios, the following schemes are introduced for the Academic Session for Class X and Class XII 2021-22.

2. Special Scheme for 2021-22 





#

#ECFGOKEUGUUKQPVQDGFKXKFGFKPVQ6GTOUYKVJCRRTQZKOCVGN[U[NNCDWUKPGCEJVGTO6JGU[NNCDWU for the Academic session 2021-22 will be divided into 2 terms by following a systematic approach by looking into the interconnectivity of concepts and topics by the Subject Experts and the Board will conduct examinations at the end of each term on the basis of thGDKHWTECVGFU[NNCDWU6JKUKUFQPGVQKPETGCUGVJGRTQDCDKNKV[QHJCXKPIC Board conducted classes X and XII examinations at the end of the academic session. $ 6JGU[NNCDWUHQTVJG$QCTFGZCOKPCVKQPYKNNDGTCVKQPCNK\GF similar to that of the last academic UGUUKQPVQDGPQVKſGFKP,WN[(QTCECFGOKEVTCPUCEVKQPUJQYGXGTUEJQQNUYKNNHQNNQYVJGEWTTKEWNWOCPF U[NNCDWUTGNGCUGFD[VJG$QCTFXKFG%KTEWNCTPQ(%$5'#ECF%WTTKEWNWOFCVGF/CTEJ5EJQQNU YKNNCNUQWUGCNVGTPCVKXGCECFGOKEECNGPFCTCPFKPRWVUHTQOVJG0%'46QPVTCPUCEVKPIVJGEWTTKEWNWO % 'HHQTVUYKNNDGOCFGVQOCMG+PVGTPCN#UUGUUOGPV2TCEVKECN2TQLGEVYQTMOQTGETGFKDNGCPFXCNKF as RGTVJGIWKFGNKPGUCPF/QFGTCVKQP2QNKE[VQDGCPPQWPEGFD[VJG$QCTFVQGPUWTGHCKTFKUVTKDWVKQPQHOCTMU

3. Details of Curriculum Transaction

   

Schools will continue teaching in distance mode till the authorities permit inperson mode of teaching in schools. %NCUUGU +:: +PVGTPCN #UUGUUOGPV VJTQWIJQWV VJG [GCTKTTGURGEVKXG QH 6GTO + CPF ++ YQWNF KPENWFG VJG  periodic tests, student enrichment, portfolio and practical work/ speaking listening activities/ project. %NCUUGU:+:+++PVGTPCN#UUGUUOGPV VJTQWIJQWVVJG[GCTKTTGURGEVKXGQH6GTO+CPF++YQWNFKPENWFGGPFQH topic or unit tests/ exploratory activities/ practicals/ projects. 5EJQQNUYQWNFETGCVGCUVWFGPVRTQſNGHQTCNNCUUGUUOGPVWPFGTVCMGPQXGTVJG[GCTCPFTGVCKPVJGGXKFGPEGUKP digital format. %$5'YKNNHCEKNKVCVGUEJQQNUVQWRNQCFOCTMUQH+PVGTPCN#UUGUUOGPVQPVJG%$5'+6RNCVHQTO Guidelines for Internal Assessment for all subjects will also be released along with the rationalized term YKUG FKXKFGF U[NNCDWU HQT VJG UGUUKQP 6JG $QCTF YQWNF CNUQ RTQXKFG CFFKVKQPCN TGUQWTEGU NKMG UCORNG assessments, question banks, teacher training etc. for more reliable and valid internal assessments.

4. Term II Examination/ Year-end Examination

At the end of the second term, the Board would organize 6GTO ++ QT ;GCTGPF 'ZCOKPCVKQP based on the TCVKQPCNK\GFU[NNCDWUQH6GTO++QPN[ KGCRRTQZKOCVGN[QHVJGGPVKTGU[NNCDWU

(iv)





6JKUGZCOKPCVKQPYQWNFDGJGNFCTQWPF/CTEJ#RTKNCVVJGGZCOKPCVKQPEGPVTGUſZGFD[VJG$QCTF 6JGRCRGTYKNNDGQHJQWTUFWTCVKQP and have questions of different formats (case-based/ situation based, open ended- short answer/ long answer type). In case the situation is not conducive for normal descriptive examination aOKPWVG/%3 based exam will be EQPFWEVGFCVVJGGPFQHVJG6GTO++CNUQ /CTMUQHVJG6GTO++'ZCOKPCVKQPYQWNFEQPVTKDWVGVQVJGſPCNQXGTCNNUEQTG

5. Assessment / Examination as per different situations    

 



#

+PECUGVJGUKVWCVKQPQHVJGRCPFGOKEKORTQXGUCPFUVWFGPVUCTGCDNGVQEQOGVQUEJQQNUQTEGPVTGU HQTVCMKPIVJGGZCOU  $QCTFYQWNFEQPFWEV6GTO+CPF6GTO++GZCOKPCVKQPUCVUEJQQNUEGPVTGUCPFVJGVJGQT[OCTMUYKNNDGFKUVTKDWVGF equally between the two exams. $ +PECUGVJGUKVWCVKQPQHVJGRCPFGOKEHQTEGUEQORNGVGENQUWTGQHUEJQQNUFWTKPI0QXGODGT&GEGODGT DWV6GTO++GZCOUCTGJGNFCVUEJQQNUQTEGPVTGU  6GTO+/%3DCUGFGZCOKPCVKQPYQWNFDGFQPGD[UVWFGPVUQPNKPGQHƀKPGHTQOJQOGKPVJKUECUGVJGYGKIJVCIG QHVJKUGZCOHQTVJGſPCNUEQTGYQWNFDGTGFWEGFCPFYGKIJVCIGQH6GTO++GZCOUYKNNDGKPETGCUGFHQTFGENCTCVKQP QHſPCNTGUWNV % +PECUGVJGUKVWCVKQPQHVJGRCPFGOKEHQTEGUEQORNGVGENQUWTGQHUEJQQNUFWTKPI/CTEJ#RTKNDWV 6GTO+GZCOUCTGJGNFCVUEJQQNUQTEGPVTGU  4GUWNVUYQWNFDGDCUGFQPVJGRGTHQTOCPEGQHUVWFGPVUQP6GTO+/%3DCUGFGZCOKPCVKQPCPFKPVGTPCNCUUGUUOGPVU 6JGYGKIJVCIGQHOCTMUQH6GTO+GZCOKPCVKQPEQPFWEVGFD[VJG$QCTFYKNNDGKPETGCUGFVQRTQXKFG[GCTGPF results of candidates. & +PECUGVJGUKVWCVKQPQHVJGRCPFGOKEHQTEGUEQORNGVGENQUWTGQHUEJQQNUCPF$QCTFEQPFWEVGF6GTO +CPF++GZCOUCTGVCMGPD[VJGECPFKFCVGUHTQOJQOGKPVJGUGUUKQP  4GUWNVUYQWNFDGEQORWVGFQPVJGDCUKUQHVJG+PVGTPCN#UUGUUOGPV2TCEVKECN2TQLGEV9QTMCPF6JGQT[OCTMU QH6GTO+CPF++GZCOUVCMGPD[VJGECPFKFCVGHTQOJQOGKP%NCUU::++UWDLGEVVQVJGOQFGTCVKQPQTQVJGT measures to ensure validity and reliability of the assessment. In all the above cases, data analysis of marks of students will be undertaken to ensure the integrity of internal assessments and home based exams.

CBSE CIRCULAR 2021-22

osQUnzh; ekè;fed f'k{kk cksMZ (f'k{kk ea=kky;] Hkkjr ljdkj osQ v/hu Lok;Ùk laxBu)

CENTRAL BOARD OF SECONDARY EDUCATION (An Autonomous Organisation Under the Ministry of Education, Govt. of India)

NO.: F.1001/CBSE-Acad/Curriculum/2021

July 22, 2021 Cir. No. A cad-53/2021

••ȱ‘Žȱ ŽŠœȱ˜ȱŒ‘˜˜•œȱŠĜ•’ŠŽȱ˜ȱ Subject : 6GTOYKUGU[NNCDWUHQT$QCTF'ZCOKPCVKQPUVQDGJGNFKPVJGCECFGOKEUGUUKQP HQT5GEQPFCT[EQPFWEVQHVJG+PVGTPCN#UUGUUOGPV2TCEVKEWO2TQLGEV 6JKUKUKPEQPVKPWCVKQPVQ$QCTFŏUEKTEWNCTPWOTGICTFKPI5RGEKCN5EJGOGQH#UUGUUOGPVHQT$QCTF'ZCOKPCVKQP HQT%NCUUGU:CPF:++HQTVJG5GUUKQP6JGUWDLGEVUHQTENCUUGU+:VQ:++CTGJGTGD[PQVKſGFXKFGU U[NNCDWUHQTVGTOGPFDQCTFGZCOKPCVKQPUIWKFGNKPGUHQTVJGEQPFWEVQH+PVGTPCN#UUGUUOGPV2TCEVKEWO2TQLGEV are also enclosed. Schools are requested to share the term wise syllabus and guidelines for the conduct of board examinations CPF+PVGTPCN#UUGUUOGPV2TCEVKEWO2TQLGEVCXCKNCDNGQP%$5'#ECFGOKE9GDUKVGhttp://cbseacademic.nic. in at the link http:/chseacademic.nicKP6GTOwise-curriculum 2022.html with all their teachers and students.

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Syllabus TERM - II Marks: 40 Units

Marks

III.

Calculus

18

IV.

Vectors and Three-Dimensional Geometry

14

VI.

Probability

8 Total

40

Internal Assessment

10

Total

50

UNIT-III: CALCULUS 1. Integrals Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the following types and problems based on them. dx 2

ax +bx+c

 )XQGDPHQWDO7KHRUHPRI&DOFXOXVZLWKRXWSURRI %DVLFSURSHUWLHVRIGH¿QLWHLQWHJUDOVDQGHYDOXDWLRQRIGH¿QLWHLQWHJUDOV 2. Applications of the Integrals 

 $SSOLFDWLRQV LQ ¿QGLQJ WKH DUHD XQGHU VLPSOH FXUYHV HVSHFLDOO\ OLQHV SDUDERODV DUHD RI FLUFOHV HOOLSVHV LQ VWDQGDUG IRUPRQO\ WKHUHJLRQVKRXOGEHFOHDUO\LGHQWL¿DEOH  3. Differential Equations



 'H¿QLWLRQRUGHUDQGGHJUHHJHQHUDODQGSDUWLFXODUVROXWLRQVRIDGLIIHUHQWLDOHTXDWLRQ6ROXWLRQRIGLIIHUHQWLDOHTXDWLRQV E\PHWKRGRIVHSDUDWLRQRIYDULDEOHVVROXWLRQVRIKRPRJHQHRXVGLIIHUHQWLDOHTXDWLRQVRI¿UVWRUGHUDQG¿UVWGHJUHHRI the type:

. Solutions of linear differential equation of the type: where p and q are functions of x or constant.

UNIT-IV: VECTORS AND THREE-DIMENSIONAL GEOMETRY 1. Vectors Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition RIYHFWRUVPXOWLSOLFDWLRQRIDYHFWRUE\DVFDODUSRVLWLRQYHFWRURIDSRLQWGLYLGLQJDOLQHVHJPHQWLQDJLYHQUDWLR'H¿QLWLRQ Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors.

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2. Three - dimensional Geometry Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, coplanar and skew lines, shortest distance between two lines. Cartesian and vector equation of a plane. Distance of a point from a plane. UNIT-VI: PROBABILITY 1. Probability Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes’ theorem, Random variable and its probability distribution. INTERNAL ASSESSMENT

10 MARKS

Periodic Test

5 Marks

0DWKHPDWLFV$FWLYLWLHV$FWLYLW\¿OHUHFRUG7HUPHQGDVVHVVPHQWRIRQHDFWLYLW\ 9LYD

5 Marks

Note: For activities NCERT Lab Manual may be referred Assessment of Activity Work: 

 ,Q ¿UVW WHUP DQ\  DFWLYLWLHV DQG LQ VHFRQG WHUP DQ\  DFWLYLWLHV VKDOO EH SHUIRUPHG E\ WKH VWXGHQW IURP WKH DFWLYLWLHV given in the NCERT Laboratory Manual for the respective class (XI or XII) which is available on the link : http://www.ncert.nic.in/exemplar/labmanuals.html a record of the same may be kept by the student. A term end test on the activity is to be conducted. The weightage are as under:



‡7KHDFWLYLWLHVSHUIRUPHGE\WKHVWXGHQWLQHDFKWHUPDQGUHFRUGNHHSLQJ



‡$VVHVVPHQWRIWKHDFWLYLW\SHUIRUPHGGXULQJWKHWHUPHQGWHVWDQG9LYDYRFH  PDUNV

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 PDUNV

CONTENTS Unit-III : CALCULUS 1. ,QGH¿QLWHDQG'H¿QLWH,QWHJUDOV ....................................................................................................................... 9 2. $UHD%HWZHHQ&XUYHV ..................................................................................................................................... 72 3. 'LIIHUHQWLDO(TXDWLRQV .................................................................................................................................. 103 UNIT-IV : VECTORS AND THREE-DIMENSIONAL GEOMETRY 4. 9HFWRU$OJHEUD .............................................................................................................................................. 134 5. 7KUHH'LPHQVLRQDO*HRPHWU\ ...................................................................................................................... 159 UNIT-VI : PROBABILITY 6. 3UREDELOLW\ .................................................................................................................................................... 185 x Sample Paper 1 (Solved) ...................................................................................................................... 218 x Sample Paper 2 (Solved) ....................................................................................................................... 228 x Sample Paper 3 (Unsolved) ................................................................................................................... 238

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1

,QGH¿QLWHDQG'H¿QLWH,QWHJUDOV

TOPICS COVERED I. INDEFINITE INTEGRALS 1. Integration of Simple Algebraic Functions and Simple Exponential Functions

2. Integration of Simple Trigonometric Functions

3. Integration by Substitution

4. Integration Using Standard Formulae

5. Directly using Formulae & Special Types of Integrals

6. Integration Using Partial Fractions

7. Integration By Parts

8. Repeating After Twice Integration

9. Integrals of Exponential Forms

10. Three More Formulae for Integration II. DEFINITE INTEGRALS

11. 'LUHFW (YDOXDWLRQ RI 'H¿QLWH ,QWHJUDOV

12. 3URSHUWLHV RI 'H¿QLWH ,QWHJUDOV

13. Odd and Even Functions

CHAPTER MAP

,,1'(),1,7(,17(*5$/6 ,QGH¿QLWH LQWHJUDO LV WKH LQYHUVHSURFHVV RI GLIIHUHQWLDWLRQ ,I GHULYDWLYH RI D IXQFWLRQ )x) is f x), then integral of function f x) LV )x  )RU H[DPSOH GHULYDWLYH RI x5 is 5x4 then integral of 5x4 will be x5 /HW XV LQWURGXFH D QHZ V\PERO ∫ f ( x ) dx  7KLV V\PERO represents integration of f x) with respect to x ,I GHULYDWLYH RI WKH IXQFWLRQ )x) is another function f x) then symbolically we write it as:

∫ f ( x ) dx

 )x  )x LV DOVR FDOOHG DQWLGHULYDWLYH RI f x 

Constant of Integration )URP WKH DERYH GH¿QLWLRQ RI LQGH¿QLWH LQWHJUDO ZH FDQ ZULWH d 4 d 4 3 4 x = 4x 3 x + 7 = 4x 3  i) Ÿ ∫ 4x dx = x  ii) dx dx

( )

( )

(

9

)

Ÿ

∫ ( 4x ) dx = x 3

4

+7

(

( )

)

 iii) d x 4 + k = 4x3 Ÿ ∫ 4x3 dx = x 4 + k dx 3 4 From the above examples, we conclude that integral of 4x3 LV QRW XQLTXH$FWXDOO\ ∫ 4x dx = x +  VRPH XQNQRZQ 4 constant) = x &,WLVWUXHIRUDOOIXQFWLRQV+HQFHZKHQZHLQWHJUDWHDIXQFWLRQZHPXVWSXWµ&¶ZLWKLWVLQWHJUDO+HUH& LVDQ\DUELWUDU\FRQVWDQWFDOOHGFRQVWDQWRILQWHJUDWLRQ In

( )

∫ f ( x ) dx = F ( x ) + C

 i) the function f x WREHLQWHJUDWHGLVFDOOHGLQWHJUDQG  ii  )x LVFDOOHGLQGH¿QLWHLQWHJUDORIf x   iii) C is called constant of integration and  iv) xLVFDOOHGYDULDEOHRILQWHJUDWLRQ

1. INTEGRATION OF SIMPLE ALGEBRAIC FUNCTIONS AND SIMPLE EXPONENTIAL FUNCTION Simple Algebraic Functions Formulae i)

⎛ x n +1 ⎞

∫ x dx = ⎜⎝ n + 1⎟⎠ + C, n ≠ − 1 n

∵

⎤ d ⎡⎛ x n +1 ⎞ + C⎥ = x n ⎢ ⎜ ⎟ dx ⎣⎝ n + 1⎠ ⎦

3

Example 1. Evaluate:

∫ x 2 dx

⎛ 3 +1 ⎞ 5 x2 ⎟ 2 + C = x2 + C I = ∫ x dx = ⎜ 5 ⎜ 3 + 1⎟ ⎝2 ⎠ 3/ 2

Solution.

 ii)

⎛ xn + 1 ⎞ n ax dx = a ⎜⎝ n + 1⎟⎠ + C, n ≠ − 1 ∫

Example 2. Evaluate:

∫ 7x

5

iii)

⎤ d ⎡ ⎛ x n +1 ⎞ a + C⎥ = ax n dx ⎢⎣ ⎜⎝ n + 1⎟⎠ ⎦

dx I=

Solution.

∵

⎡ x5+1 ⎤

∫ 7x dx = 7 ⎢⎣ 5 + 1⎥⎦ + C 5

Ÿ I = 7 x6 + C 6

∫ ⎡⎣ f ( x ) + g(x) + φ (x)⎤⎦ dx = ∫ f ( x ) dx + ∫ g(x) dx + ∫ φ(x) dx + C

Example 3. Evaluate:

1⎞ ⎛ 5 2 2 dx x + 3 x + 5 x ∫ ⎜⎝ ⎟ ⎠

I=

Solution.

⎛

∫ ⎜⎝ x

5

1⎞ 1 + 3x 2 + 5x 2 ⎟ dx = ∫ x5 dx + ∫ 3x 2 dx + ∫ 5x 2 dx ⎠

⎛ 3⎞ ⎛ x3 ⎞ x6 x2 1 10 3/ 2 + 3 ⎜ ⎟ + 5 ⎜ ⎟ + C = x 6 + x3 + x +C I= 6 6 3 ⎜ 3⎟ ⎝ 3⎠ ⎝ 2⎠

( )

iv)

∫ k dx = kx + C, Here k is constant

Example 4. Evaluate:

v)

10

∫ (ax + b) dx = n

d [ kx + C] = k dx

∵

n +1 ⎤ n d ⎡ ( ax + b ) + C⎥ = ( ax + b ) ⎢ dx ⎢ a ( n + 1) ⎥⎦ ⎣

∫ 4 dx I=

Solution.

∵

∫ 4 dx

= 4x + C

(ax + b) n + 1 + C, n ≠ − 1 a(n + 1)

MATHEMATICS–12

Example 5. Evaluate:

∫ (3x + 5)

7

I=

Solution. Example 6. Evaluate:

dx 3x + 5) (3x + 5) + C = 1 (3x + 5)8 + C +C = 24 24 (7 + 1) 7 +1

( 7 ∫ (3x + 5) dx = 3

8

dx 8x + 3 + 8x + 1

∫

dx 8x + 3 + 8x + 1

∫

I=

Solution.

Rationalise denominators

∫

I=

(

)

8x + 3 − 8x + 1 dx 8x + 3 − 8x − 1

=∫

(

2

⎡ 3/ 2 3/ 2 ⎤ 8x + 3) 8x + 1) ⎥ ( ( 1 1 ⎡ ⎢ I= +C= − (8x + 3)3/ 2 − (8x + 1)3/ 2 ⎤⎦ + C 3 ×8 ⎥ 2 ⎢ 3 ×8 24 ⎣ ⎢⎣ 2 ⎥⎦ 2

Ÿ

Example 7. Evaluate:

∫

(

x2 −

)

1 ⎛ 1⎞ ⎜ x + 3 ⎟⎠ dx x ⎝ x

(

x2 −

⎛

3

Solution.

I=

∫

Ÿ

I=

∫ ⎜⎝ x

Example 8. Evaluate:

∫

(x

2

+

)

+ 2 (3x + 5) x2

(x

2

)

1 ⎛ 1⎞ ⎜ x + 3 ⎟⎠ dx x ⎝ x 1 1 1 x4 + log x − x + 3 + C − 1 − 4 ⎞⎟ dx = 4 x x ⎠ 3x dx

)

+ 2 (3x + 5)

Solution.

I=

∫

Ÿ

I=

∫ ⎜⎝

Ÿ

2 I = 3x + 5x + 6 log x − 10 + C 2 x

Example 9. Evaluate:

x2

dx

⎛ 3x3 + 5x 2 + 6x + 10 ⎞ 6 10 ⎞ ⎛ ⎟⎠ dx = ∫ ⎜⎝ 3x + 5 + x + x 2 ⎟⎠ dx x2

⎡ ( x − 3) ( x + 2) ( x + 7 ) ⎤ ⎥ dx ( x − 4) ⎦

∫ ⎢⎣

I=

Solution.

⎡ ( x − 3) ( x + 2) ( x + 7 ) ⎤ ⎥ dx ( x − 4) ⎦

∫ ⎢⎣

(

⎡ ( x − 3) x 2 + 9x + 14 ⎢ I= ∫ x−4 ⎢ ⎣

Ÿ

I= ?

)

8x + 3 − 8x + 1 dx

) ⎥⎤ dx = ⎥ ⎦

⎛ x3 + 9x 2 + 14x − 3x 2 − 27x − 42 ⎞ ⎟⎠ dx ∫ ⎜⎝ x−4

⎡ x3 + 6x 2 − 13x − 42 ⎤ ⎥ dx ( x − 4) ⎦

∫ ⎢⎣

(

)

x3 + 6x 2 − 13x − 42 = 2 66 x + 10x + 27 + x−4 x−4 I=

∫ ⎢⎣( x ⎡

2

)

+ 10x + 27 +

66 ⎤ dx x − 4 ⎥⎦

3 = x + 5x 2 + 27x + 66 log x − 4 + C 3

vi)

dx

−1

∫ x n = (n − 1) x n − 1 + C, n ≠ 1

∵

⎤ ⎡ d ⎢ −1 ⎥ = 1n + C dx ⎢ ( n − 1) x n −1 ⎥ x ⎣ ⎦

( )

INDEFINITE AND DEFINITE INTEGRALS 11

Example 10. Evaluate:

1

∫ x 4 dx I=

Solution.

vii)

−1

⎛ 1⎞

−1

∫ ⎜⎝ x 4 ⎟⎠ dx = ( 4 − 1) x 4−1 + C = 3x3 + C

−1

dx

∫ (ax + b)n = a (n − 1) (ax + b)n − 1 + C, n ≠ 1

Example 11. Evaluate:

∵

⎤ ⎪⎫ d ⎡⎢ ⎪⎧ −1 1 ⎥ + C = ⎨ ⎬ n dx ⎢ ⎪ a ( n − 1) ( ax + b )n −1 ⎪ ax b) + ⎥ ( ⎭ ⎣⎩ ⎦

dx

∫ ( 4x + 3)5 I=

Solution.

−1

dx

∫ ( 4x + 3)5 = 4 (5 − 1) ( 4x + 3)5−1 + C

⎤ ⎤ ⎡ ⎡ −1 −1 I= ⎢ 4⎥ +C = ⎢ 4⎥ +C ⎢⎣ 4 × 4 ( 4x + 3) ⎥⎦ ⎢⎣16 ( 4x + 3) ⎥⎦ viii)

dx = log | x | + C x

∫

Example 12. Evaluate:

∫

ix)

⎛ dx ⎞

∫

1

d ⎡1 1 log ax + b + C⎤ = ⎥⎦ ax + b dx ⎣⎢ a

dx

∫ (7x + 9) I=

Solution. Example 14. Evaluate:

∵

dx = log x + C x

∫ ⎜⎝ ax + b ⎟⎠ = a log | ax + b | + C

Example 13. Evaluate:

d 1 ⎡log x + C⎤⎦ = dx ⎣ x

dx x I=

Solution.

∵

∫

1

7x + 9 + C = 1 log 7x + 9 + C 7

(3x − 7)2 dx 3x + 2 I=

Solution.

dx

∫ (7x + 9) = 7 log

∫

(3x − 7)2 dx 3x + 2

⎛ 9x 2 − 42x + 49 ⎞ I = ∫⎜ ⎟⎠ dx 3x + 2 ⎝

Ÿ

9x 2 − 42x + 49 81 = (3x − 16) + 3x + 2 3x + 2

?

⎡

∫ ⎢⎣(3x − 16) +

(

81 3x + 2

)

⎤ 3x 2 81 ⎥⎦ dx = 2 − 16x + 3 log 3x + 2 + C

?

I=

Ÿ

2 I = 3x − 16x + 27 log 3x + 2 + C 2

x)

⎛

∫ ⎜⎝

dx ⎞ 2 ax + b = +C a ax + b ⎟⎠

Example 15. Evaluate: Solution.

12

∫

∵

dx 5x − 3 I=

∫

dx 2 5x − 3 = +C 5 5x − 3

MATHEMATICS–12

⎤ d ⎡ 2 ax + b + C⎥ = ⎢ dx ⎣ a ⎦

1 ax + b

Simple Exponential Functions i)

∫e

x

 ii)

∫e

mx + n



dx = e x + C dx =

1 mx + n e +C m

Example 16. Evaluate:

∫e

4x + 7

I=

Solution.

 iii)

dx

∫e

4x + 7

dx =

I= mx + n

dx =

5x

∫ 5 dx = log 5 + C x

a mx + n +C m log a

Example 18. Evaluate:

∫7

Ÿ I=

I=

1 4x + 7 e +C 4

⎤ d ⎡ ax + C⎥ = a x dx ⎢⎣ log a ⎦ 5x +C log 5

Ÿ I=

⎤ d ⎡ a mx + n + C⎥ = a mx + n dx ⎢⎣ m log a ⎦

∵

( 2x + 5) ∫ 7 dx =

7( 2x + 5) +C 2 log 7

( 2x+ 5) Ÿ I= 7 +C 2 log 7

⎛ 5 x + 23x ⎞ dx 7 2x ⎟⎠

∫ ⎜⎝

I=

Solution.

⎛ 5 x + 23x ⎞ ⎡ 5 x 23x ⎤ 2x ⎟ dx = ∫ ⎢ 2x + 2x ⎥ dx ⎠ 7 7 ⎦ ⎣7

∫ ⎜⎝

5 8 ( 49 ) + ( 49 ) + C 8 ( ) + ( 49 ) ⎥⎦ dx = log ( 5 ) log ( 8 ) 49 49

⎡ ( 5) x (8)x ⎤⎥ dx = ⎡ 5 + I= ∫⎢ ∫ ⎢⎣ 49 x x ⎢⎣ ( 49) ( 49) ⎥⎦ Example 20. Evaluate:

()

( 2x + 5)dx

Solution. Example 19. Evaluate:

d ⎡ 1 mx + n e + C⎤ = e mx + n ⎥⎦ dx ⎢⎣ m

∵

Solution.

∫a

∵

1 4x + 7 e +C 4

ax

Example 17. Evaluate: ∫ 5 xdx

 iv)

d ⎡ x e + C⎤⎦ = e x dx ⎣

()

∫ a dx = log a + C x

∵

∫ (2

x

)

x

x

x⎤

+ 3x dx

I=

Solution.

2

x

=

∫ (2

x

)

+ 3x dx = ∫ ⎡( 2) + (3) + 2 × 2( x) × 3( x) ⎤ dx ⎣ ⎦ 2

2x

2x

4x 9x 2.6 x x x x⎤ ⎡ + + +C 4 + 9 + 2 × 6 dx = ∫⎣ ⎦ log 4 log 9 log 6

Exercise 1.1 Evaluate each of the following Integrals Simple Algebraic Functions I. Very Short Answer Type Questions 1.

∫ (ax

4.

∫ x7

7.

∫ 9x + 5

10.

2

+ bx + c) dx

dx dx

⎛

∫ ⎜⎝

(1 Mark) ⎤ ⎡ 3 2. ∫ ⎢5x 2 + x + 7 ⎥ dx ⎣ ⎦ dx 5. ∫ (4x + 7)5 8.

∫ (3x

3

+ 5) 2 dx

3.

∫ (7x + 5)

6.

∫

dx 7x − 3

9.

∫x

2

5

dx

⎡ x 4 + x + 1 + 3⎤ dx x ⎦⎥ ⎣⎢

2

x−

1 ⎞ ⎟ dx x⎠

INDEFINITE AND DEFINITE INTEGRALS 13

II. Short Answer Type Questions 11.

∫ ⎡⎣5

14.

∫

(2 Marks)

x + 7x3 x + x x ⎤⎦ dx

( ) x+

1 ⎛ 2 1 ⎞ x + ⎟ dx x ⎜⎝ x⎠

dx 4x + 3 + 4x − 5

12.

∫

15.

∫ ⎜⎝ x +

⎛

1 ⎞ ⎟ x⎠

(

III. Long Answer Type Questions-I

x+

(

∫ (x + 1) (x + 2) (x + 3) dx

18.

∫x

20.

⎡ (3x + 2) (2x − 3) 2 ⎤ ⎥ dx ∫ ⎢⎣ x2 ⎦

21.

⎡ (2x + 1) 2 ⎤ ∫ ⎢⎣ 3x + 2 ⎥⎦ dx

23.

⎡ x 4 + x 2 + 1⎤ ∫ ⎢⎣ x 2 + x + 1 ⎥⎦ dx

24.

⎡ x4 + 7 ⎤ ∫ ⎢⎣ x + 1 ⎥⎦ dx

∫

16.

∫

dx 8 − 5x − 7 − 5x

( ) x+

1 ⎛ 2 1⎞ ⎜ x − 3 ⎟⎠ dx x ⎝ x (3 Marks)

)

⎛ 1 − x 2 ⎞ x3 + 1 dx ⎜⎝ 2 ⎟⎠ x x

17.

2

)

1 dx x

13.

(x + 1) (3x − 1) dx x2 2

19.

∫

22.

∫ ⎢⎣

⎡ (x + 1) (x + 2) (x + 3) ⎤ ⎥ dx (x + 4) ⎦

Simple Exponential Functions I. Very Short Answer Type Questions (4x + 3)

25.

∫e

28.

x 2x ∫ (5 × 3 ) dx

dx

(1 Mark)

26.

∫7

29.

∫ ⎜⎝

(2x − 5)

dx

⎛ 8 x + 3x ⎞ dx 5 x ⎟⎠

27.

∫ (3

30.

∫ ⎜⎝

x

× 2 x) dx

⎛ ax + bx ⎞ dx a xb x ⎟⎠

31.

∫ (e

3 log x

)

+ e 2 log x dx

II. Long Answer Type Question-I 32.

(

⎡ 3x + 5 ∫ ⎢⎢ 22x ⎢⎣ 3

) ⎥ dx ⎥ ⎥⎦

$QVZHUVDQG+LQWV

2

ax bx cx + + +D 3 2 (7x + 5)6 +C 3. 42 1.

5.

(3 Marks)

2x 2 ⎤

−1 +C 16(4x + 7) 4

2.

5 2x 2

+ x + 7x + C 2

−1 +C 6x 6 2 7x − 3 6. +C 7

8.

9x 7 + 15x 4 + 25x + C 7 2 Hint:([SDQGx2 + 5)2DQGLQWHJUDWH

5 2 9. x + x + log | x | + 3x + C 5 2 Hint:)LUVWPXOWLSO\DQGWKHQLQWHJUDWH

x2 − 2x + log | x | + C 2 2 Hint: Expand ⎛⎜ x − 1 ⎞⎟ DQGWKHQLQWHJUDWH ⎝ x⎠ 3 9 5 11. 10 x 2 + 14 x 2 + 2 x 2 + C 3 9 5 3 3⎤ ⎡ 12. 1 ⎢(4x + 3) 2 − (4x − 5) 2 ⎥ + C 48 ⎣ ⎦ 10.

dx 4x + 3 + 4 x − 5   5DWLRQDOLVH'UDQGLQWHJUDWH 3 3⎤ ⎡ 13. − 2 ⎢(8 − 5x) 2 + (7 − 5x) 2 ⎥ + C 15 ⎣ ⎦ Hint:

∫

Hint:5DWLRQDOLVHWKH'UDQGWKHQLQWHJUDWH

14

14.

4.

7. 1 log | 9x + 5 | + C 9

MATHEMATICS–12

−1

3

2

x4 2 x 2 x2 + + − 2x 2 + C 4 3 2 Hint:0XOWLSO\WZRIDFWRUVDQGWKHQLQWHJUDWH −1

5

15.

2 x 2 + 2x − 2x 2 + C 5 Hint:0XOWLSO\WZRIDFWRUVDQGWKHQLQWHJUDWH

4 2 16. x + 1 + x + 1 3 + C 4 x 2 3x Hint:)LUVWPXOWLSO\WZRIDFWRUVDQGWKHQLQWHJUDWH 4 2 17. x + 2x3 + 11x + 6x + C 4 2 Hint:)LUVWPXOWLSO\DOOWKHIDFWRUVDQGWKHQLQWHJUDWH 8

x 18. log | x | − + C 8 Hint:)LUVWPXOWLSO\IDFWRUVDQGWKHQLQWHJUDWH 3x 2 + 3x − log | x | + 1 + C 19. 2 x

( x + 1)(3x 2 − 1)

= 3x + 3 − 1 − 12   1RZLQWHJUDWH x x x 18 2 20. 6x − 28x + 3 log | x | − + C x Hint: Expand Nr and divide by x2WKHQLQWHJUDWH Hint:

21.

2

2x 2 + 4x + 1 log | 3x + 2 | + C 3 9 27 Hint:

( 2x + 1)2 3x + 2

4x 2 + 4x + 1 3x + 2 4 4 1 = x+ + 3 9 9 (3x + 2) =

22.

x3 x 2 + + 3x − 6 log | x + 4 | + C 3

27.

6x + C Hint: 3x × 2x = 6x log 6

( x + 1)( x + 2)( x + 3) ( x + 4)

28.

(45) x + C Hint: 5x × 32x = 5x × 9x  x log (45)

Hint:

=

x3 + 6x 2 + 11x + 6 ( x + 4)

2 = x + 2x + 3 −

23.

x3 x 2 x − + +C 3 2

(

)(

8 3 ( ) ( ) +C 5 5 29. + log ( 8 ) log ( 3 ) 5 5 8 +3 8 3 Hint: = ( ) +( ) 5 5 5 x

6

( x + 4)

)

x2 + x + 1 x2 − x + 1 x4 + x2 + 1 = x2 + x + 1 x2 + x + 1

Hint:

(

)

x

 1RZLQWHJUDWH −1 1 30. − +C (a x) log a (b x) log b

 1RZLQWHJUDWH

4 3 2 24. x − x + x − x + 8 log | x + 1 | + C 4 3 2

(

ax + bx 1 1 = x + x 1RZLQWHJUDWH a xb x b a 4 3 ⎤ ⎡ m log x log x m 31. x + x + C; =e = x m ⎦⎥ ⎢∵ e ⎣ 4 3 Hint:

)

x4 − 1 + 8 x4 + 7 = x +1 x +1 =

( )

( x + 1)( x − 1)( x 2 + 1)

(

x +1

)

3

+ 8 x +1

log x + elog x Hint: e3 log x + e2 log x = e

= ( x − 1) x 2 + 1 + 8 x +1 8 3 2 = x − x + x −1+ x +1 

32.

( 32 )

(2x − 5)

26. 7 +C 2 log 7

2x

x

+ 52x 2

2x

)

2

=

x

32x + 54x + 2 ( 75) 22x

 1RZLQWHJUDWH

x

() ( ) ( )

= 3 2 

2

25 2 ( 75 ) ( ) 2 4 + + +C 3 25 2 log ( ) 2 log ( ) log ( 75 ) 2 2 4 2x

(3 Hint:

 1RZLQWHJUDWH

25. 1 ⎡⎣e(4x + 3) ⎤⎦ + C 4

x



= x – x

Hint:

x

x

x

2



x

2x

+ 25 2

2x

+ 2 75 4

x

2. INTEGRATION OF SIMPLE TRIGONOMETRIC FUNCTIONS Formulae d ( − cos x ) + C = sin x dx d (sin x + C) = cos x dx d ⎡log sec x + C⎤⎦ = tan x dx ⎣ d ⎡log sin x + C⎤⎦ = cot x dx ⎣ d ⎡log sec x + tan x + C⎤⎦ = sec x dx ⎣

1.

∫ sin x dx = − cos x + C

∵

2.

∫ cos x dx = sin x + C

∵

3.

∫ tan x dx = log | sec x | + C

∵

4.

∫ cot x dx = log | sin x | + C

∵

5.

∫ sec x dx = log | sec x + tan x | + C

∵

6.

∫ cosec x dx = log | cosec x − cot x | + C = − log | cosec x + cot x | + C

∵ d ⎡⎣log ( cosec x − cot x ) + C⎤⎦ = cosec x dx

or

7.

∫ sec

8.

∫ cosec x dx = − cot x + C

∵

9.

∫ (sec x tan x) dx = sec x + C

∵

∫ (cosec x cot x) dx = − cosec x + C

∵

10.

2

x dx = tan x + C 2

∵

d ⎡ − log cosec x + cot x + C⎤⎦ = cosec x dx ⎣ d [ tan x + C] = sec2 x dx d [ − cot x + C] = cosec2 x dx d (sec x ) + C = sec x tan x dx d [ −cosec x ] + C = cosec x cot x dx

INDEFINITE AND DEFINITE INTEGRALS 15

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